JEE Advanced 2016 · Paper 1 · Mathematics Q48

← JEE Advanced 2016 · Paper 1

MathematicsMultiple correct+4 / -2Question 48 of 54

The circle $C_1:x^2+y^2=3$ , with centre at $O$ , intersects the parabola $x^2=2y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_1$ at $P$ touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$ , respectively. Suppose $C_2$ and $C_3$ have equal radii $2\sqrt3$ and centres $Q_2$ and $Q_3$ , respectively. If $Q_2$ and $Q_3$ lie on the $y$ -axis, then

Q_2Q_3=12

R_2R_3=4\sqrt6

area of the triangle

OR_2R_3

6\sqrt2

area of the triangle

PQ_2Q_3

4\sqrt2

Answer and worked solution

Official answer: A, B, C

Point P. $x^2+y^2=3$ and $x^2=2y$ give $y^2+2y-3=0$ , so $y=1$ and $P=(\sqrt2,1)$ .

Tangent at P. $\sqrt2x+y=3$ .

Centres. $Q=(0,b)$ with $\dfrac{|b-3|}{\sqrt3}=2\sqrt3$ , so $b=9$ or $-3$ .

(A) $Q_2Q_3=12$ . True.
(B) $R_2R_3$ is the projection of $Q_2Q_3$ on the tangent, whose direction is $(1,-\sqrt2)/\sqrt3$ : $12\cdot\dfrac{\sqrt2}{\sqrt3}=4\sqrt6$ . True.
(C) Area of $OR_2R_3=\frac12\cdot4\sqrt6\cdot\dfrac3{\sqrt3}=6\sqrt2$ . True.
(D) Area of $PQ_2Q_3=\frac12\cdot12\cdot\sqrt2=6\sqrt2$ , not $4\sqrt2$ . False.

(A), (B), (C)

Topics:Tangents and normals Parabola

← Previous question Next question →

JEE Advanced 2016 · Paper 1, Mathematics question 48