Let the line L:√2x+y=α pass through the point of the intersection P(in the first quadrant)of the circle x^2+y^2=3 and the parabola x^2=2y. Let the…
JEE Main 2024 — Mathematics Coordinate Geometry
2024integerhard
Let the line L:2x+y=α pass through the point of the intersection P(in the first quadrant)of the circle x2+y2=3 and the parabola x2=2y. Let the line L touch two circles C1 and C2 of equal radius 23. If the centres Q1 and Q2 of the circles C1 and C2 lie on the y−axis, then the square of the area of the triangle PQ1Q2 is equal to _________.
Official previous-year question
Held on 1 Feb 2024 · Verified 6 Jul 2026.
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Solution
Given: x2+y2=3 and x2=2y
⇒y2+2y−3=0
⇒(y+3)(y−1)=0
⇒y=1,−3(Rejected as it gives imaginary value ofx)
⇒y=1
⇒x=2
⇒P≡(2,1)
Now, P lies on the line 2x+y=α.
⇒2(2)+1=α
⇒α=3
For circle C1, Q1 lies on y−axis.
Let Q1≡(0,β) and R1=23 (given)
Line L act as tangent so applying the condition of tangency
⇒P=r
⇒∣3β−3∣=23
⇒∣β−3∣=6
⇒β−3=6,−6
⇒β=9,−3
So, {Q}_{1}\equiv (0,9)&{Q}_{2}\equiv (0,-3)
⇒Ar(ΔPQ1Q2)=21∣20019−3111∣
⇒Ar(ΔPQ1Q2)=21(2(12))
⇒Ar(ΔPQ1Q2)=62
⇒(ΔPQ1Q2)2=72
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