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Ellipse

25 questions · Mathematics · JEE Advanced

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JEE Advanced
Mathematics
Ellipse

Ellipse

25 questions · Mathematics · JEE Advanced

Let $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2})$ be two distinct points on the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ such that $y_{1} > 0$ , and $y_{2} > 0$ . Let $C$ denote the circle $x^{2} + y^{2} = 9$ , and $M$ be the point $(3, 0)$ .…2025 · Shift 2 · Q23 · Multiple correct
Consider the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ . Let $S (p, q)$ be a point in the first quadrant such that $\frac{p ^{2}}{9} + \frac{q ^{2}}{4} > 1$ . Two tangents are drawn from $S$ to the ellipse, of which one meets the ellipse at one end…2024 · Shift 1 · Q21 · MCQ
Let $T_{1}$ and $T_{2}$ be two distinct common tangents to the ellipse $E : \frac{x ^{2}}{6} + \frac{y ^{2}}{3} = 1$ and the parabola $P : y^{2} = 12 x$ . Suppose that the tangent $T_{1}$ touches $P$ and $E$ at the points $A_{1}$ and $A_{2}$ , respectively and the…2023 · Shift 1 · Q19 · Multiple correct
Consider the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{3} = 1$ Let $H (α, 0), 0 < α < 2$ , be a point. A straight line drawn through $H$ parallel to the $y$ -axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ … Includes table2022 · Shift 1 · Q36 · MCQ
Let E be the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1$ . For any three distinct points P, Q and Q' on E, let M(P, Q) be the mid-point of the line segment joining P and Q, and M(P, Q') be the mid-point of the line segment…2021 · Shift 2 · Q37 · Numerical
Define the collections {E1, E2, E3, ...} of ellipses and {R1, R2, R3.....} of rectangles as follows : $E_{1} : \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ R1 : rectangle of largest area, with sides parallel to the axes, inscribed in E1; En :…2019 · Shift 1 · Q24 · Multiple correct
Let S be the circle in the XY-plane defined the equation x2 + y2 = 4. Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point…2018 · Shift 1 · Q34 · MCQ
Consider two straight lines, each of which is tangent to both the circle x2 + y2 = (1/2) and the parabola y2 = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre is at the origin O(0, 0) and whose semi-major…2018 · Shift 2 · Q22 · Multiple correct
For how many values of p, the circle x2 + y2 + 2x + 4y $-$ p = 0 and the coordinate axes have exactly three common points?2017 · Shift 1 · Q29 · Numerical
Let $F_{1} (x_{1}, 0)$ and $F_{2} (x_{2}, 0)$ for $x_{1} < 0$ and $x_{2} > 0$ , be the foci of the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{8} = 1$ . Suppose a parabola having vertex at the origin and…2016 · Shift 2 · Q29 · MCQ
Let $F_{1} (x_{1}, 0)$ and $F_{2} (x_{2}, 0)$ for $x_{1} < 0$ and $x_{2} > 0$ , be the foci of the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{8} = 1$ . Suppose a parabola having vertex at the origin and…2016 · Shift 2 · Q30 · MCQ
Let $E_{1}$ and $E_{2}$ be two ellipses whose centres are at the origin. The major axes of $E_{1}$ and $E_{2}$ lie along the $x$ -axis and the $y$ -axis, respectively. Let $S$ be the circle $x^{2} + (y - 1)^{2} = 2$ . The…2015 · Shift 2 · Q29 · Multiple correct
The common tangents to the circle $x^{2} + y^{2} = 2$ and the parabola $y^{2} = 8 x$ touch the circle at the points $R$ and the parabola at the points $R$ , $S$ . Then the area of the quadrilateral $P QR S$ is2014 · Shift 2 · Q31 · MCQ
A vertical line passing through the point $(h, 0)$ intersects the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{3} = 1$ at the points $P$ and $Q$ . Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$ . If $Δ (h)$ …2013 · Shift 1 · Q30 · Numerical
The ellipse $E_{1} : \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_{2}$ passing through the point $(0, 4)$ circumscribes the rectangle $R$ . The…2012 · Shift 1 · Q27 · MCQ
Tangents are drawn from the point $P (3, 4)$ to the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ touching the ellipse at points $A$ and $B$ . The coordinates of $A$ and $B$ are2010 · Shift 2 · Q27 · MCQ
Tangents are drawn from the point $P (3, 4)$ to the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ touching the ellipse at points $A$ and $B$ . The orthocentre of the triangle $P A B$ is2010 · Shift 2 · Q28 · MCQ
Tangents are drawn from the point $P (3, 4)$ to the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ touching the ellipse at points $A$ and $B$ . The equation of the locus of the point whose distances from the point $P$ and the line $A B$ …2010 · Shift 2 · Q29 · MCQ
Match the conics in Column I with the statements/expressions in Column II : Includes table2009 · Shift 1 · Q30 · MCQ
In a triangle $A B C$ with fixed base $B C$ , the vertex $A$ moves such that $cos B + cos C = 4 sin^{2} \frac{A}{2} .$ If $c$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $C$ and $C$ , respectively,…2009 · Shift 1 · Q31 · Multiple correct
The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse $x^{2} + 9 y^{2} = 9$ meets its auxiliary circle at the point $M$ . Then the area of the triangle with vertices at $A$ , $M$ and…2009 · Shift 1 · Q32 · MCQ
The normal at a point $P$ on the ellipse $x^{2} + 4 y^{2} = 16$ meets the $x$ - axis $Q$ . If $M$ is the mid point of the line segment $P Q$ , then the locus of $M$ intersects the latus rectums of the given ellipse at the points2009 · Shift 2 · Q34 · MCQ
An ellipse intersects the hyperbola $2 x^{2} - 2 y^{2} = 1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes then2009 · Shift 2 · Q36 · Multiple correct
Consider the two curves $C_{1} : y^{2} = 4 x, C_{2} : x^{2} + y^{2} - 6 x + 1 = 0$ . Then,2008 · Shift 1 · Q34 · MCQ
Let $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2}), y_{1} < 0, y_{2} < 0,$ be the end points of the latus rectum of the ellipse $x^{2} + 4 y^{2} = 4.$ The equations of parabolas with latus rectum $P Q$ are :2008 · Shift 1 · Q41 · Multiple correct