The circle C1:x2+y2=3 having centre at origin,O intersects the parabola x2=2y at the point P in the first quadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively. Suppose C2 and C3 have equal radii 2√ and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the Y-axis then
area of the ΔOR2R3 is 6√
Given C1:x2+y2=3 intersects the parabola x2=2y
On solving x2+y2=3 and x2=2y, we get
∴ y=1,−3 [neglecting y=−3,as−√≤y≤√]
⇒ P(√,1)ϵ I quadrant
Equation of tangent at P(√,1) to C1:x2+y2=3 is
Now, let the centres of C2 and C3 be Q2 and Q3, and tangent at P touches C2 and C3 at R2 and R3 shown as below
Let Q2 be (0, k) and radius is 2√.
∴ |0+k−3|√=2√ (Distance of centre Q2 from tangent line)⇒ |k−3|=6⇒ k=9,−3∴ Q2(0,9) and Q3(0,−3)
∴ Option (a) is correct.
Also R2R3 is common internal tangent to C2 and C3
∴ Option (b) is correct
∵ Length of perpendicular from O(0, 0) to R2R3 is equal to radius of C1=√.
∴Area of ΔOR2R3=12×R2R3×√=12×4√×√=6√
∴ Option (c) is correct.
Also are ΔPQ2Q3=12Q2Q3×√=√2×12=6√
∴ Option (d) is incorrect