Straight Lines and Pair of Straight Lines

152 questions · Mathematics · JEE Main

Straight Lines and Pair of Straight Lines

152 questions · Mathematics · JEE Main

Let the area of the triangle formed by a straight line $L : x + b y + c = 0$ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of $4 5^{\circ}$ with the…2025 · 2 Apr · Shift 2 · Q37 · MCQ
A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $L_{1} : 2 x + y + 6 = 0$ and $L_{2} : 4 x + 2 y - p = 0, p > 0$ , at the points A and B , respectively. If $A B = \frac{9}{2}$ …2025 · 3 Apr · Shift 1 · Q44 · MCQ
Consider the lines $x (3 λ + 1) + y (7 λ + 2) = 17 λ + 5, λ$ being a parameter, all passing through a point P. One of these lines (say $L$ ) is farthest from the origin. If the distance of $L$ from the point $(3, 6)$ is $d$ , then…2025 · 3 Apr · Shift 2 · Q32 · MCQ
Let the three sides of a triangle are on the lines $4 x - 7 y + 10 = 0, x + y = 5$ and $7 x + 4 y = 15$ . Then the distance of its orthocentre from the orthocentre of the tringle formed by the lines $x = 0, y = 0$ and $x + y = 1$ is2025 · 4 Apr · Shift 1 · Q43 · MCQ
Let ABC be the triangle such that the equations of lines AB and AC be $3 y - x = 2$ and $x + y = 2$ , respectively, and the points B and C lie on $x$ -axis. If P is the orthocentre of the triangle ABC , then the area of the triangle PBC is equal to2025 · 7 Apr · Shift 1 · Q30 · MCQ
If the orthocenter of the triangle formed by the lines y = x + 1, y = 4x - 8 and y = mx + c is at (3, -1), then m - c is :2025 · 7 Apr · Shift 2 · Q45 · MCQ
Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle $α$ with the positive x-axis and the equations of its diagonals are $(3 + 1) x + (3 - 1) y = 0$ and $(3 - 1) x - (3 + 1) y + 83 = 0$ …2025 · 8 Apr · Shift 2 · Q29 · MCQ
A line passing through the point P( $a$ , 0) makes an acute angle $α$ with the positive x-axis. Let this line be rotated about the point P through an angle $\frac{α}{2}$ in the clockwise direction. If in the new position, the slope…2025 · 8 Apr · Shift 2 · Q30 · MCQ
Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2 y = 2$ . If the centroid of $△ PQR$ is the point $(α, β)$ , then $15 (α - β)$ is equal to :2025 · 22 Jan · Shift 1 · Q30 · MCQ
Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $P QR$ is formed such that $Q$ lies on one of the parallel lines, while R lies on…2025 · 22 Jan · Shift 2 · Q49 · Numerical
A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x - y + 2 = 0$ and $y + 2 = 0$ , respectively. If the locus of the point $P$ , that divides the rod $A B$ internally in the ratio $2 : 1$ is $9 (x^{2} + α y^{2} + β x y + γ x + 28 y) - 76 = 0$ …2025 · 23 Jan · Shift 2 · Q41 · MCQ
Let the lines $3 x - 4 y - α = 0, 8 x - 11 y - 33 = 0$ , and $2 x - 3 y + λ = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2 x - 3 y + λ = 0$ is $(\frac{57}{13}, \frac{- 40}{13})$ , then $∣ α λ ∣$ is equal to2025 · 24 Jan · Shift 1 · Q43 · MCQ
Let the points $(\frac{11}{2}, α)$ lie on or inside the triangle with sides $x + y = 11, x + 2 y = 16$ and $2 x + 3 y = 29$ . Then the product of the smallest and the largest values of $α$ is equal to :2025 · 24 Jan · Shift 2 · Q40 · MCQ
Two equal sides of an isosceles triangle are along $- x + 2 y = 4$ and $x + y = 4$ . If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$ , is:2025 · 28 Jan · Shift 2 · Q31 · MCQ
If A and B are the points of intersection of the circle $x^{2} + y^{2} - 8 x = 0$ and the hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ and a point P moves on the line $2 x - 3 y + 4 = 0$ , then the centroid of $Δ P A B$ lies on the line :2025 · 28 Jan · Shift 2 · Q37 · MCQ
Let ΔABC be a triangle formed by the lines 7x – 6y + 3 = 0, x + 2y – 31 = 0 and 9x – 2y – 19 = 0. Let the point (h, k) be the image of the centroid of ΔABC in the line 3x + 6y – 53 = 0. Then h2 + k2 + hk is equal to :2025 · 29 Jan · Shift 1 · Q32 · MCQ
Let the line x + y = 1 meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of…2025 · 29 Jan · Shift 2 · Q33 · MCQ
Let $A B C$ be an isosceles triangle in which $A$ is at $(- 1, 0), ∠ A = \frac{2 π}{3}, A B = A C$ and $B$ is on the positve $x$ -axis. If $BC = 43$ and the line $BC$ intersects the line $y = x + 3$ at $(\alpha,…2024 · 1 Feb · Shift 2 · Q56 · Numerical
The lines $L_{1}, L_{2}, \dots, L_{20}$ are distinct. For $n = 1, 2, 3, \dots, 10$ all the lines $L_{2 n - 1}$ are parallel to each other and all the lines $L_{2 n}$ pass through a given point…2024 · 1 Feb · Shift 2 · Q60 · Numerical
The vertices of a triangle are $A (- 1, 3), B (- 2, 2)$ and $C (3, - 1)$ . A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to…2024 · 4 Apr · Shift 1 · Q41 · MCQ
Let two straight lines drawn from the origin $O$ intersect the line $3 x + 4 y = 12$ at the points $P$ and $Q$ such that $△ OPQ$ is an isosceles triangle and $∠ POQ = 9 0^{\circ}$ . If $l = OP^{2} + PQ^{2} + QO^{2}$ …2024 · 5 Apr · Shift 1 · Q33 · MCQ
Let $A (- 1, 1)$ and $B (2, 3)$ be two points and $P$ be a variable point above the line $AB$ such that the area of $△ PAB$ is 10. If the locus of $P$ is $\mathrm{a}…2024 · 5 Apr · Shift 2 · Q38 · MCQ
Let a variable line of slope $m > 0$ passing through the point $(4, - 9)$ intersect the coordinate axes at the points $A$ and $B$ . The minimum value of the sum of the distances of $A$ and $B$ from the origin is2024 · 6 Apr · Shift 1 · Q32 · MCQ
If the locus of the point, whose distances from the point $(2, 1)$ and $(1, 3)$ are in the ratio $5 : 4$ , is $a x^{2} + b y^{2} + c x y + d x + ey + 170 = 0$ , then the value of $a^{2} + 2 b + 3 c + 4 d + e$ is equal to :2024 · 6 Apr · Shift 2 · Q49 · MCQ
The equations of two sides $AB$ and $AC$ of a triangle $ABC$ are $4 x + y = 14$ and $3 x - 2 y = 5$ , respectively. The point $(2, - \frac{4}{3})$ divides the third side $BC$ internally in the ratio $2 : 1$ …2024 · 8 Apr · Shift 1 · Q45 · MCQ
If the orthocentre of the triangle formed by the lines $2 x + 3 y - 1 = 0, x + 2 y - 1 = 0$ and $a x + b y - 1 = 0$ , is the centroid of another triangle, whose circumcentre and orthocentre respectively are $(3, 4)$ and $(- 6, - 8)$ , then the value of $∣ a - b ∣$ is…2024 · 8 Apr · Shift 1 · Q51 · Numerical
If the line segment joining the points $(5, 2)$ and $(2, a)$ subtends an angle $\frac{π}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is :2024 · 8 Apr · Shift 2 · Q36 · MCQ
Let a ray of light passing through the point $(3, 10)$ reflects on the line $2 x + y = 6$ and the reflected ray passes through the point $(7, 2)$ . If the equation of the incident ray is $a x + b y + 1 = 0$ , then $a^{2} + b^{2} + 3 ab$ is equal to …2024 · 8 Apr · Shift 2 · Q60 · Numerical
A variable line $L$ passes through the point $(3, 5)$ and intersects the positive coordinate axes at the points $A$ and $B$ . The minimum area of the triangle $OAB$ , where $O$ is the origin, is :2024 · 9 Apr · Shift 1 · Q45 · MCQ
A ray of light coming from the point $P (1, 2)$ gets reflected from the point $Q$ on the $x$ -axis and then passes through the point $R (4, 3)$ . If the point $S (h, k)$ is such that $P QR S$ is a parallelogram, then $h k^{2}$ is…2024 · 9 Apr · Shift 1 · Q48 · MCQ
The portion of the line $4 x + 5 y = 20$ in the first quadrant is trisected by the lines $L_{1}$ and $L_{2}$ passing through the origin. The tangent of an angle between the lines $L_{1}$ and $L_{2}$ is :2024 · 27 Jan · Shift 1 · Q44 · MCQ
Let $R$ be the interior region between the lines $3 x - y + 1 = 0$ and $x + 2 y - 5 = 0$ containing the origin. The set of all values of $a$ , for which the points $(a^{2}, a + 1)$ lie in $R$ , is :2024 · 27 Jan · Shift 2 · Q45 · MCQ
If the sum of squares of all real values of $α$ , for which the lines $2 x - y + 3 = 0, 6 x + 3 y + 1 = 0$ and $α x + 2 y - 2 = 0$ do not form a triangle is $p$ , then the greatest integer less than or equal to $p$ is .2024 · 27 Jan · Shift 2 · Q56 · Numerical
In a $△ A B C$ , suppose $y = x$ is the equation of the bisector of the angle $B$ and the equation of the side $A C$ is $2 x - y = 2$ . If $2 A B = B C$ and the points $A$ and $B$ are respectively $(4, 6)$ and $(α, β)$ , then $α + 2 β$ …2024 · 29 Jan · Shift 1 · Q42 · MCQ
Let $A$ be the point of intersection of the lines $3 x + 2 y = 14, 5 x - y = 6$ and $B$ be the point of intersection of the lines $4 x + 3 y = 8, 6 x + y = 5$ . The distance of the point $P (5, - 2)$ from the line $AB$ is2024 · 29 Jan · Shift 2 · Q32 · MCQ
The distance of the point $(2, 3)$ from the line $2 x - 3 y + 28 = 0$ , measured parallel to the line $3 x - y + 1 = 0$ , is equal to2024 · 29 Jan · Shift 2 · Q34 · MCQ
A line passing through the point $A (9, 0)$ makes an angle of $3 0^{\circ}$ with the positive direction of $x$ -axis. If this line is rotated about A through an angle of $1 5^{\circ}$ in the clockwise direction, then its equation in…2024 · 30 Jan · Shift 1 · Q38 · MCQ
If $x^{2} - y^{2} + 2 h x y + 2 g x + 2 f y + c = 0$ is the locus of a point, which moves such that it is always equidistant from the lines $x + 2 y + 7 = 0$ and $2 x - y + 8 = 0$ , then the value of $g + c + h - f$ equals2024 · 30 Jan · Shift 2 · Q38 · MCQ
Let $α, β, γ, δ \in Z$ and let $A (α, β), B (1, 0), C (γ, δ)$ and $D (1, 2)$ be the vertices of a parallelogram $ABCD$ . If $A B = 10$ and the points $A$ and $C$ lie…2024 · 31 Jan · Shift 1 · Q34 · MCQ
Let $A (a, b), B (3, 4)$ and $C (- 6, - 8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P (2 a + 3, 7 b + 5)$ from the line $2 x + 3 y - 4 = 0$ measured parallel to the line $x - 2 y - 1 = 0$ is2024 · 31 Jan · Shift 2 · Q49 · MCQ
Let $A (- 2, - 1), B (1, 0), C (α, β)$ and $D (γ, δ)$ be the vertices of a parallelogram $A B C D$ . If the point $C$ lies on $2 x - y = 5$ and the point $D$ lies on $3 x - 2 y = 6$ , then the value of $∣ α + β + γ + δ ∣$ is…2024 · 31 Jan · Shift 2 · Q54 · Numerical
If the orthocentre of the triangle, whose vertices are (1, 2), (2, 3) and (3, 1) is $(α, β)$ , then the quadratic equation whose roots are $α + 4 β$ and $4 α + β$ , is :2023 · 1 Feb · Shift 1 · Q33 · MCQ
The straight lines $l_{1}$ and $l_{2}$ pass through the origin and trisect the line segment of the line L : $9 x + 5 y = 45$ between the axes. If $m_{1}$ and $m_{2}$ are the slopes of the lines $l_{1}$ …2023 · 6 Apr · Shift 1 · Q25 · MCQ
Let $C (α, β)$ be the circumcenter of the triangle formed by the lines $4 x + 3 y = 694 y - 3 x = 17$ , and $x + 7 y = 61$ . Then $(α - β)^{2} + α + β$ is equal to :2023 · 8 Apr · Shift 1 · Q31 · MCQ
Let the equations of two adjacent sides of a parallelogram $ABCD$ be $2 x - 3 y = - 23$ and $5 x + 4 y = 23$ . If the equation of its one diagonal $AC$ is $3 x + 7 y = 23$ and the distance of A from the other diagonal is $d$ ,…2023 · 10 Apr · Shift 2 · Q38 · Numerical
If the point $(α, \frac{7 3}{3})$ lies on the curve traced by the mid-points of the line segments of the lines $x cos θ + y sin θ = 7, θ \in (0, \frac{π}{2})$ between the co-ordinates…2023 · 12 Apr · Shift 1 · Q29 · MCQ
Let $(α, β)$ be the centroid of the triangle formed by the lines $15 x - y = 82, 6 x - 5 y = - 4$ and $9 x + 4 y = 17$ . Then $α + 2 β$ and $2 α - β$ are the roots of the equation :2023 · 13 Apr · Shift 2 · Q29 · MCQ
If $(α, β)$ is the orthocenter of the triangle $ABC$ with vertices $A (3, - 7), B (- 1, 2)$ and $C (4, 5)$ , then $9 α - 6 β + 60$ is equal to :2023 · 15 Apr · Shift 1 · Q34 · MCQ
The equations of the sides AB, BC and CA of a triangle ABC are : $2 x + y = 0, x + p y = 21 a, (a \pm 0)$ and $x - y = 3$ respectively. Let P(2, a) be the centroid of $Δ$ ABC. Then (BC) $^{2}$ is equal to .2023 · 24 Jan · Shift 2 · Q41 · Numerical
Let $B$ and $C$ be the two points on the line $y + x = 0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y - 2 x = 2$ such that $△ A B C$ is an equilateral triangle. Then, the area of the $△ A B C$ …2023 · 29 Jan · Shift 1 · Q23 · MCQ
A light ray emits from the origin making an angle 30 $^{\circ}$ with the positive $x$ -axis. After getting reflected by the line $x + y = 1$ , if this ray intersects $x$ -axis at Q, then the abscissa of Q is :2023 · 29 Jan · Shift 1 · Q30 · MCQ
Let $A (\frac{3}{a}, a), a > 0$ , be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C. If $D (3 cos θ, a sin θ)$ is a point in the fourth quadrant…2022 · 24 Jun · Shift 1 · Q37 · Numerical
Let the area of the triangle with vertices A(1, $α$ ), B( $α$ , 0) and C(0, $α$ ) be 4 sq. units. If the points ( $α$ , $- α$ ), ( $- α$ , $α$ ) and ( $α$ 2, $β$ ) are collinear, then $β$ is equal to :2022 · 24 Jun · Shift 2 · Q30 · MCQ
A line, with the slope greater than one, passes through the point $A (4, 3)$ and intersects the line $x - y - 2 = 0$ at the point B. If the length of the line segment $A B$ is $\frac{29}{3}$ , then $B$ also lies on the line :2022 · 25 Jul · Shift 1 · Q32 · MCQ
Let the point $P (α, β)$ be at a unit distance from each of the two lines $L_{1} : 3 x - 4 y + 12 = 0$ , and $L_{2} : 8 x + 6 y + 11 = 0$ . If $P$ lies below $L_{1}$ and above $L_{2}$ , then $100 (α + β)$ is equal to :2022 · 25 Jul · Shift 2 · Q30 · MCQ
A point $P$ moves so that the sum of squares of its distances from the points $(1, 2)$ and $(- 2, 1)$ is 14. Let $f (x, y) = 0$ be the locus of $P$ , which intersects the $x$ -axis at the points $A$ , $B$ and the $y$ -axis…2022 · 26 Jul · Shift 1 · Q34 · MCQ
The equations of the sides $AB, BC$ and $CA$ of a triangle $ABC$ are $2 x + y = 0, x + p y = 15 a$ and $x - y = 3$ respectively. If its orthocentre is $(2, a), - \frac{1}{2} < a < 2$ , then…2022 · 26 Jul · Shift 1 · Q43 · Numerical
Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle. Then the area of $Δ$ PQR is :2022 · 26 Jun · Shift 1 · Q29 · MCQ
Let $A (1, 1), B (- 4, 3), C (- 2, - 5)$ be vertices of a triangle $A B C, P$ be a point on side $B C$ , and $Δ_{1}$ and $Δ_{2}$ be the areas of triangles $A P B$ and $A B C$ , respectively. If $Δ_{1} : Δ_{2} = 4 : 7$ , then the area…2022 · 27 Jul · Shift 1 · Q36 · MCQ
The equations of the sides $AB, BC$ and CA of a triangle ABC are $2 x + y = 0, x + p y = 39$ and $x - y = 3$ respectively and $P (2, 3)$ is its circumcentre. Then which of the following is NOT true?2022 · 27 Jul · Shift 2 · Q31 · MCQ
In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie on the line x + 3y = 7. If ( $α$ , $β$ ) is the centroid of $Δ$ ABC, then 15( $α$ + $β$ ) is equal to :2022 · 27 Jun · Shift 1 · Q29 · MCQ
A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2 : 1. Let the…2022 · 28 Jun · Shift 1 · Q39 · Numerical
Let a triangle be bounded by the lines L1 : 2x + 5y = 10; L2 : $-$ 4x + 3y = 12 and the line L3, which passes through the point P(2, 3), intersects L2 at A and L1 at B. If the point P divides the line-segment AB, internally in the ratio 1…2022 · 28 Jun · Shift 2 · Q33 · MCQ
Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1,1). If the line AP intersects the line BC at the point Q $(k_{1}, k_{2})$ , then $k_{1} + k_{2}$ is equal to :2022 · 29 Jul · Shift 1 · Q33 · MCQ
Let $m_{1}, m_{2}$ be the slopes of two adjacent sides of a square of side a such that $a^{2} + 11 a + 3 (m_{1}^{2} + m_{2}^{2}) = 220$ . If one vertex of the square is $(10 (cos α - sin α), 10 (sin α + cos α))$ ,…2022 · 29 Jul · Shift 2 · Q31 · MCQ
Let $A (α, - 2), B (α, 6)$ and $C (\frac{α}{4}, - 2)$ be vertices of a $△ ABC$ . If $(5, \frac{α}{4})$ is the circumcentre of $△ ABC$ , then…2022 · 29 Jul · Shift 2 · Q32 · MCQ
The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A' B (where B is the point (2, 3)) subtend angle $\frac{π}{4}$ at the origin, is equal to :2022 · 29 Jun · Shift 1 · Q29 · MCQ
The distance of the origin from the centroid of the triangle whose two sides have the equations $x - 2 y + 1 = 0$ and $2 x - y - 1 = 0$ and whose orthocenter is $(\frac{7}{3}, \frac{7}{3})$ is :2022 · 29 Jun · Shift 2 · Q31 · MCQ
Let $α$ 1, $α$ 2 ( $α$ 1 2) be the values of $α$ fo the points ( $α$ , $-$ 3), (2, 0) and (1, $α$ ) to be collinear. Then the equation of the line, passing through ( $α$ 1, $α$ 2) and making an angle of $\frac{π}{3}$ …2022 · 30 Jun · Shift 1 · Q31 · MCQ
Let the points of intersections of the lines x $-$ y + 1 = 0, x $-$ 2y + 3 = 0 and 2x $-$ 5y + 11 = 0 are the mid points of the sides of a triangle $Δ$ ABC. Then, the area of the $Δ$ ABC is .2021 · 1 Sep · Shift 2 · Q41 · Numerical
A man starts walking from the point P( $-$ 3, 4), touches the x-axis at R, and then turns to reach at the point Q(0, 2). The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then $50 ((P R)^{2} + (R Q)^{2})$ …2021 · 1 Sep · Shift 2 · Q46 · Numerical
Let A( $-$ 1, 1), B(3, 4) and C(2, 0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let A1 and A2 be the areas of $Δ$ ABC and $Δ$ PQC respectively, such that A1 = 3A2,…2021 · 16 Mar · Shift 2 · Q38 · MCQ
In a triangle PQR, the co-ordinates of the points P and Q are ( $-$ 2, 4) and (4, $-$ 2) respectively. If the equation of the perpendicular bisector of PR is 2x $-$ y + 2 = 0, then the centre of the circumcircle of the $Δ$ PQR is :2021 · 17 Mar · Shift 1 · Q23 · MCQ
The maximum value of z in the following equation z = 6xy + y2, where 3x + 4y $\leq$ 100 and 4x + 3y $\leq$ 75 for x $\geq$ 0 and y $\geq$ 0 is .2021 · 17 Mar · Shift 1 · Q39 · Numerical
Let tan $α$ , tan $β$ and tan $γ$ ; $α$ , $β$ , $γ e \frac{( 2 n - 1 ) π}{2}$ , n $\in$ N be the slopes of three line segments OA, OB and OC, respectively, where O is origin. If circumcentre of $Δ$ ABC coincides…2021 · 17 Mar · Shift 2 · Q36 · Numerical
The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :2021 · 18 Mar · Shift 1 · Q25 · MCQ
The equation of one of the straight lines which passes through the point (1, 3) and makes an angles $tan^{- 1} (2)$ with the straight line, y + 1 = 3 $2$ x is :2021 · 18 Mar · Shift 1 · Q29 · MCQ
A square ABCD has all its vertices on the curve x2y2 = 1. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is .2021 · 18 Mar · Shift 1 · Q46 · Numerical
Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of $Δ$ ABC,…2021 · 18 Mar · Shift 2 · Q34 · MCQ
Consider a triangle having vertices A( $-$ 2, 3), B(1, 9) and C(3, 8). If a line L passing through the circum-centre of triangle ABC, bisects line BC, and intersects y-axis at point $(0, \frac{α}{2})$ , then the value of…2021 · 20 Jul · Shift 2 · Q42 · Numerical
A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$ . Three stones A, B and C are placed at the points (1, 1), (2, 2) and (4, 4) respectively.…2021 · 24 Feb · Shift 1 · Q30 · MCQ
The image of the point (3, 5) in the line x $-$ y + 1 = 0, lies on :2021 · 25 Feb · Shift 1 · Q36 · MCQ
The intersection of three lines x $-$ y = 0, x + 2y = 3 and 2x + y = 6 is a :2021 · 26 Feb · Shift 1 · Q30 · MCQ
Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :2021 · 27 Aug · Shift 1 · Q23 · MCQ
The point P (a, b) undergoes the following three transformations successively : (a) reflection about the line y = x. (b) translation through 2 units along the positive direction of x-axis. (c) rotation through angle $\frac{π}{4}$ about…2021 · 27 Jul · Shift 2 · Q23 · MCQ
Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point :2021 · 27 Jul · Shift 2 · Q32 · MCQ
If p and q are the lengths of the perpendiculars from the origin on the lines, x cosec $α -$ y sec $α$ = k cot 2 $α$ and x sin $α$ + y cos $α$ = k sin2 $α$ respectively, then k2 is equal to :2021 · 31 Aug · Shift 1 · Q29 · MCQ
Let A be the set of all points ( $α$ , $β$ ) such that the area of triangle formed by the points (5, 6), (3, 2) and ( $α$ , $β$ ) is 12 square units. Then the least possible length of a line segment joining the origin to a…2021 · 31 Aug · Shift 2 · Q33 · MCQ
The set of all possible values of $θ$ in the interval (0, $π$ ) for which the points (1, 2) and (sin $θ$ , cos $θ$ ) lie on the same side of the line x + y = 1 is :2020 · 2 Sep · Shift 2 · Q34 · MCQ
If a $Δ$ ABC has vertices A(–1, 7), B(–7, 1) and C(5, –5), then its orthocentre has coordinates :2020 · 3 Sep · Shift 2 · Q35 · MCQ
Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m) above the line AC is :2020 · 4 Sep · Shift 1 · Q28 · MCQ
If the perpendicular bisector of the line segment joining the points P(1 ,4) and Q(k, 3) has y-intercept equal to –4, then a value of k is :2020 · 4 Sep · Shift 2 · Q30 · MCQ
If the line, 2x - y + 3 = 0 is at a distance $\frac{1}{5}$ and $\frac{2}{5}$ from the lines 4x - 2y + $α$ = 0 and 6x - 3y + $β$ = 0, respectively, then the sum of all possible values of $α$ and $β$ is :2020 · 5 Sep · Shift 1 · Q21 · Numerical
A ray of light coming from the point (2, $23$ ) is incident at an angle 30o on the line x = 1 at the point A. The ray gets reflected on the line x = 1 and meets x-axis at the point B. Then, the line AB passes through the point :2020 · 6 Sep · Shift 1 · Q35 · MCQ
Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (–1, –4) in this line is :2020 · 6 Sep · Shift 2 · Q19 · MCQ
Let A(1, 0), B(6, 2) and C $(\frac{3}{2}, 6)$ be the vertices of a triangle ABC. If P is a Point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ,…2020 · 7 Jan · Shift 1 · Q32 · Numerical
The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line x = y is :2020 · 7 Jan · Shift 2 · Q25 · MCQ
Let two points be A(1, –1) and B(0, 2). If a point P(x', y') be such that the area of $Δ$ PAB = 5 sq. units and it lies on the line, 3x + y – 4 $λ$ = 0, then a value of $λ$ is :2020 · 8 Jan · Shift 1 · Q34 · MCQ
Let C be the centroid of the triangle with vertices (3, –1), (1, 3) and (2, 4). Let P be the point of intersection of the lines x + 3y – 1 = 0 and 3x – y + 1 = 0. Then the line passing through the points C and P also passes through the…2020 · 9 Jan · Shift 1 · Q37 · MCQ
Let O(0, 0) and A(0, 1) be two fixed points. Then the locus of a point P such that the perimeter of $Δ$ AOP is 4, is :2019 · 8 Apr · Shift 1 · Q30 · MCQ
A point on the straight line, 3x + 5y = 15 which is equidistant from the coordinate axes will lie only in :2019 · 8 Apr · Shift 1 · Q36 · MCQ
If the system of linear equations x – 2y + kz = 1 2x + y + z = 2 3x – y – kz = 3 has a solution (x,y,z), z $e$ 0, then (x,y) lies on the straight line whose equation is :2019 · 8 Apr · Shift 2 · Q37 · MCQ
Suppose that the points (h,k), (1,2) and (–3,4) lie on the line L1 . If a line L2 passing through the points (h,k) and (4,3) is perpendicular to L1 , then $\frac{k}{h}$ equals :2019 · 8 Apr · Shift 2 · Q44 · MCQ
Slope of a line passing through P(2, 3) and intersecting the line, x + y = 7 at a distance of 4 units from P, is :2019 · 9 Apr · Shift 1 · Q44 · MCQ
If the two lines x + (a – 1) y = 1 and 2x + a2y = 1 (a $\in$ R – {0, 1}) are perpendicular, then the distance of their point of intersection from the origin is :2019 · 9 Apr · Shift 2 · Q39 · MCQ
Let the equations of two sides of a triangle be 3x $-$ 2y + 6 = 0 and 4x + 5y $-$ 20 = 0. If the orthocentre of this triangle is at (1, 1), then the equation of its third side is :2019 · 9 Jan · Shift 2 · Q35 · MCQ
The region represented by| x – y | $\leq$ 2 and | x + y| $\leq$ 2 is bounded by a :2019 · 10 Apr · Shift 1 · Q30 · MCQ
Lines are drawn parallel to the line 4x – 3y + 2 = 0, at a distance $\frac{3}{5}$ from the origin. Then which one of the following points lies on any of these lines ?2019 · 10 Apr · Shift 2 · Q33 · MCQ
If the line 3x + 4y – 24 = 0 intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is :2019 · 10 Jan · Shift 1 · Q34 · MCQ
A point P moves on the line 2x – 3y + 4 = 0. If Q(1, 4) and R (3, – 2) are fixed points, then the locus of the centroid of $Δ$ PQR is a line :2019 · 10 Jan · Shift 1 · Q39 · MCQ
If 5, 5r, 5r2 are the lengths of the sides of a triangle, then r cannot be equal to :2019 · 10 Jan · Shift 1 · Q44 · MCQ
Two vertices of a triangle are (0, 2) and (4, 3). If its orthocenter is at the origin, then its third vertex lies in which quadrant :2019 · 10 Jan · Shift 2 · Q35 · MCQ
Two sides of a parallelogram are along the lines, x + y = 3 & x – y + 3 = 0. If its diagonals intersect at (2, 4), then one of its vertex is :2019 · 10 Jan · Shift 2 · Q41 · MCQ
If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5), then the equation of the diagonal AD is :2019 · 11 Jan · Shift 2 · Q40 · MCQ
A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of 60o with the line x + y = 0. Then an equation of the line L is…2019 · 12 Apr · Shift 2 · Q28 · MCQ
If the straight line, 2x – 3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, $β$ ), then $β$ equals :2019 · 12 Jan · Shift 1 · Q42 · MCQ
If a straight line passing through the point P(–3, 4) is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is :2019 · 12 Jan · Shift 2 · Q35 · MCQ
In a triangle ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are respectively, x + y = 5 and x = 4. Then area of $Δ$ ABC (in sq. units) is :2018 · 15 Apr · Shift 1 · Q40 · MCQ
The foot of the perpendicular drawn from the origin, on the line, 3x + y = $λ$ ( $λ e$ 0) is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is :2018 · 15 Apr · Shift 2 · Q35 · MCQ
The sides of a rhombus ABCD are parallel to the lines, x $-$ y + 2 = 0 and 7x $-$ y + 3 = 0. If the diagonals of the rhombus intersect P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the coordinate of A is :2018 · 15 Apr · Shift 2 · Q36 · MCQ
A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is :2018 · Shift 0 · Q37 · MCQ
A square, of each side 2, lies above the x-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle 30o with the positive direction of the x-axis, then the sum of the x-coordinates of the…2017 · 9 Apr · Shift 1 · Q43 · MCQ
Let k be an integer such that the triangle with vertices (k, – 3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point :2017 · Shift 0 · Q36 · MCQ
The point (2, 1) is translated parallel to the line L : x− y = 4 by $23$ units. If the newpoint Q lies in the third quadrant, then the equation of the line passing through Q and perpendicular to L is :2016 · 9 Apr · Shift 1 · Q38 · MCQ
If a variable line drawn through the intersection of the lines $\frac{x}{3} + \frac{y}{4} = 1$ and $\frac{x}{4} + \frac{y}{3} = 1,$ meets the coordinate axes at A and B, (A $e$ B), then the locus of the midpoint of AB is :2016 · 9 Apr · Shift 1 · Q40 · MCQ
A straight line through origin O meets the lines 3y = 10 − 4x and 8x + 6y + 5 = 0 at points A and B respectively. Then O divides the segment AB in the ratio :2016 · 10 Apr · Shift 1 · Q36 · MCQ
A ray of light is incident along a line which meets another line, 7x − y + 1 = 0, at the point (0, 1). The ray is then reflected from this point along the line, y + 2x = 1. Then the equation of the line of incidence of the ray of light is :2016 · 10 Apr · Shift 1 · Q37 · MCQ
Two sides of a rhombus are along the lines, $x - y + 1 = 0$ and $7 x - y - 5 = 0$ . If its diagonals intersect at $(- 1, - 2)$ , then which one of the following is a vertex of this rhombus?2016 · Shift 0 · Q33 · MCQ
The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0) (0, 41)$ and $(41, 0)$ is :2015 · Shift 0 · Q38 · MCQ
Let $a, b, c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4 a x + 2 a y + c = 0$ and $5 b x + 2 b y + d = 0$ lies in the fourth quadrant and is equidistant from the two axes then :2014 · Shift 0 · Q37 · MCQ
Let $P S$ be the median of the triangle with vertices $P (2, 2)$ , $Q (6, - 1)$ and $R (7, 3)$ . The equation of the line passing through $(1, - 1)$ band parallel to PS is :2014 · Shift 0 · Q38 · MCQ
The $x$ -coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $(0, 1) (1, 1)$ and $(1, 0)$ is :2013 · Shift 0 · Q37 · MCQ
A ray of light along $x + 3 y = 3$ gets reflected upon reaching $X$ -axis, the equation of the reflected ray is :2013 · Shift 0 · Q38 · MCQ
If the line $2 x + y = k$ passes through the point which divides the line segment joining the points $(1, 1)$ and $(2, 4)$ in the ratio $3 : 2$ , then $k$ equals :2012 · Shift 0 · Q39 · MCQ
The line $L$ given by $\frac{x}{5} + \frac{y}{b} = 1$ passes through the point $(13, 32)$ . The line K is parrallel to $L$ and has the equation $\frac{x}{c} + \frac{y}{3} = 1.$ Then the distance between $L$ and $K$ is :2010 · Shift 0 · Q43 · MCQ
The lines $p (p^{2} + 1) x - y + q = 0$ and $(p^{2} + 1)^{2} x + (p^{2} + 1) y + 2 q$ $= 0$ are perpendicular to a common line for :2009 · Shift 0 · Q40 · MCQ
The shortest distance between the line $y - x = 1$ and the curve $x = y^{2}$ is :2009 · Shift 0 · Q41 · MCQ
The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept -4. Then a possible value of k is :2008 · Shift 0 · Q47 · MCQ
Let A $(h, k)$ , B $(1, 1)$ and C $(2, 1)$ be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is $1$ square unit, then the set of values which $^{'} k^{'}$ can take is…2007 · Shift 0 · Q56 · MCQ
A straight line through the point $A (3, 4)$ is such that its intercept between the axes is bisected at $A$ . Its equation is :2006 · Shift 0 · Q76 · MCQ
If $(a, a^{2})$ falls inside the angle made by the lines $y = \frac{x}{2}, x > 0$ and $y = 3 x, x > 0,$ then a belong to :2006 · Shift 0 · Q77 · MCQ
The line parallel to the $x$ - axis and passing through the intersection of the lines $a x + 2 b y + 3 b = 0$ and $b x - 2 a y - 3 a = 0,$ where $(a, b) e (0, 0)$ is :2005 · Shift 0 · Q103 · MCQ
If non zero numbers $a, b, c$ are in $H . P .,$ then the straight line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point. That point is :2005 · Shift 0 · Q113 · MCQ
If a vertex of a triangle is $(1, 1)$ and the mid points of two sides through this vertex are $(- 1, 2)$ and $(3, 2)$ then the centroid of the triangle is :2005 · Shift 0 · Q93 · MCQ
The equation of the straight line passing through the point $(4, 3)$ and making intercepts on the co-ordinate axes whose sum is $- 1$ is :2004 · Shift 0 · Q107 · MCQ
Let $A (2, - 3)$ and $B (- 2, 1)$ be vertices of a triangle $A B C$ . If the centroid of this triangle moves on the line $2 x + 3 y = 1$ , then the locus of the vertex $C$ is the line :2004 · Shift 0 · Q108 · MCQ
A square of side a lies above the $x$ -axis and has one vertex at the origin. The side passing through the origin makes an angle $α (0 < α < \frac{π}{4})$ with the positive direction of x-axis. The equation…2003 · Shift 0 · Q109 · MCQ
Locus of centroid of the triangle whose vertices are $(a cos t, a sin t), (b sin t, - b cos t)$ and $(1, 0),$ where $t$ is a parameter, is :2003 · Shift 0 · Q110 · MCQ
If $x_{1}, x_{2}, x_{3}$ and $y_{1}, y_{2}, y_{3}$ are both in G.P. with the same common ratio, then the points $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ :2003 · Shift 0 · Q111 · MCQ
If the equation of the locus of a point equidistant from the point $(a_{1,} b_{1})$ and $(a_{2,} b_{2})$ is $(a_{1} - a_{2}) x + (b_{1} - b_{2}) y + c = 0$ , then the value…2003 · Shift 0 · Q112 · MCQ
A triangle with vertices $(4, 0), (- 1, - 1), (3, 5)$ is :2002 · Shift 0 · Q72 · MCQ
Locus of mid point of the portion between the axes of $x cos α + y sin α = p$ where $p$ is constant is :2002 · Shift 0 · Q73 · MCQ