Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Straight Lines and Pair of Straight Lines question

2022 · 24 Jun · Shift 2 · Q30
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Straight Lines and Pair of Straight Lines
  5. /2022 · 24 Jun · Shift 2 · Q30

Straight Lines and Pair of Straight Lines question

2022 · 24 Jun · Shift 2 · Q30

JEE MainMathematicsStraight Lines and Pair of Straight LinesMCQ+4 / −1
Let the area of the triangle with vertices A(1, α\alphaα), B(α\alphaα, 0) and C(0, α\alphaα) be 4 sq. units. If the points (α\alphaα, −α-\alpha−α), (−α-\alpha−α, α\alphaα) and (α\alphaα 2, β\betaβ) are collinear, then β\betaβ is equal to :
  1. A
    64
  2. B
    −-− 8
  3. C
    −-− 64
  4. D
    512
View written solutionFree

Correct answer: C

  1. Use the area condition to find α\alphaα

The vertices are: A(1,α),B(α,0),C(0,α)A(1,\alpha),\quad B(\alpha,0),\quad C(0,\alpha)A(1,α),B(α,0),C(0,α)

Using the coordinate area formula,

Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\text{Area} = \frac12 \left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|Area=21​∣x1​(y2​−y3​)+x2​(y3​−y1​)+x3​(y1​−y2​)∣

Substitute:

Area=12∣1(0−α)+α(α−α)+0(α−0)∣=12∣−α∣=∣α∣2\text{Area} = \frac12 \left|1(0-\alpha)+\alpha(\alpha-\alpha)+0(\alpha-0)\right| = \frac12 |-\alpha| = \frac{|\alpha|}{2}Area=21​∣1(0−α)+α(α−α)+0(α−0)∣=21​∣−α∣=2∣α∣​

Given area =4=4=4,

∣α∣2=4  ⟹  ∣α∣=8\frac{|\alpha|}{2}=4 \implies |\alpha|=82∣α∣​=4⟹∣α∣=8

So,

α=8orα=−8\alpha=8 \quad \text{or} \quad \alpha=-8α=8orα=−8
  1. Use the collinearity condition

The three points are:

P(α,−α),Q(−α,α),R(α2,β)P(\alpha,-\alpha),\quad Q(-\alpha,\alpha),\quad R(\alpha^2,\beta)P(α,−α),Q(−α,α),R(α2,β)

First find the line through PPP and QQQ.

Slope of PQPQPQ:

m=α−(−α)−α−α=2α−2α=−1m=\frac{\alpha-(-\alpha)}{-\alpha-\alpha} = \frac{2\alpha}{-2\alpha}=-1m=−α−αα−(−α)​=−2α2α​=−1

(for α≠0\alpha\neq 0α=0, which is true here)

So the line through P(α,−α)P(\alpha,-\alpha)P(α,−α) with slope −1-1−1 is:

y+α=−1(x−α)y+\alpha = -1(x-\alpha)y+α=−1(x−α) y+α=−x+αy+\alpha = -x+\alphay+α=−x+α y=−xy=-xy=−x

Hence any collinear point on this line must satisfy:

β=−α2\beta = -\alpha^2β=−α2
  1. Substitute possible values of α\alphaα

Since ∣α∣=8|\alpha|=8∣α∣=8,

α2=64\alpha^2 = 64α2=64

Thus,

β=−α2=−64\beta = -\alpha^2 = -64β=−α2=−64
  1. Check with options

The correct option is:

−64\boxed{-64}−64​

which is Option C.

  1. Compare with stored correct answer

Stored correct answer: C

Our derived answer: C

So they agree.

PreviousNext

More from Straight Lines and Pair of Straight Lines

  • 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 AB is 329​​, then B also lies on the line :2022 · MCQ
  • Let the point P(α,β) be at a unit distance from each of the two lines L1​:3x−4y+12=0, and L2​:8x+6y+11=0. If P lies below L1​ and above L2​, then 100(α+β) is equal to :2022 · 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 · MCQ
  • The equations of the sides AB,BC and CA of a triangle ABC are 2x+y=0,x+py=15a and x−y=3 respectively. If its orthocentre is (2,a),−21​<a<2, then…2022 · 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 · MCQ
  • Let A(1,1),B(−4,3),C(−2,−5) be vertices of a triangle ABC,P be a point on side BC, and Δ1​ and Δ2​ be the areas of triangles APB and ABC, respectively. If Δ1​:Δ2​=4:7, then the area…2022 · MCQ
  • The equations of the sides AB,BC and CA of a triangle ABC are 2x+y=0,x+py=39 and x−y=3 respectively and P(2,3) is its circumcentre. Then which of the following is NOT true?2022 · 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 · MCQ