The circle $C_{1}: x^{2}+y^{2}=3$, with centre at $O$, intersects the parabola $x^{2}=2 y$ 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 $2 \sqrt{3}$ and centres $Q_{2}$ and $Q_{3}$, respectively. If $Q_{2}$ and $Q_{3}$ lie on the $y$-axis, then

(A) $Q_{2} Q_{3}=12$; (B) $R_{2} R_{3}=4 \sqrt{6}$; (C) area of the triangle $OR_{2} R_{3}$ is $6 \sqrt{2}$

(D) area of the triangle $P Q_{2} Q_{3}$ is $4 \sqrt{2}$

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Circle

The circle

C_{1} : x^{2} + y^{2} = 3

, with centre at

O

, intersects the parabola

x^{2} = 2 y

at the point

P

in the first quadrant. Let the tangent to the circle

C_{1}

P

touches other two circles

C_{2}

and

C_{3}

R_{2}

and

R_{3}

, respectively. Suppose

C_{2}

and

C_{3}

have equal radii

23

and centres

Q_{2}

and

Q_{3}

, respectively. If

Q_{2}

and

Q_{3}

lie on the

y

-axis, then

(A) $Q_{2} Q_{3} = 12$ ✓ Correct
(B) $R_{2} R_{3} = 46$ ✓ Correct
(C)area of the triangle $O R_{2} R_{3}$ is $62$ ✓ Correct
(D)area of the triangle $P Q_{2} Q_{3}$ is $42$

Answer: (A), (B), (C)

Solution

Solving

x^{2} + y^{2} = 3

and

x^{2} = 2 y

gives

2 y + y^{2} = 3

, so

y^{2} + 2 y - 3 = 0

, thus

y = 1

(first quadrant). Then

x = 2

, so

P (2, 1)

.

Tangent at

P

to circle:

2 x + y = 3

.

Centres

Q_{2} (0, α_{2})

and

Q_{3} (0, α_{3})

y

-axis with radius

23

.

Distance from centre to tangent equals radius:

\frac{∣ α - 3∣}{3} = 23 \Rightarrow ∣ α - 3∣ = 6

α = 9

α = - 3

. Thus

Q_{2} (0, - 3)

and

Q_{3} (0, 9)

.

(A)

Q_{2} Q_{3} = 12

. True.

(B) Length of transverse common tangent:

R_{2} R_{3} = (Q_{2} Q_{3})^{2} - (r_{2} + r_{3})^{2} = 144 - 48 = 96 = 46

. True.

(C) Perpendicular distance from

O

to tangent:

\frac{3}{3} = 3

. Area

= \frac{1}{2} \cdot 46 \cdot 3 = 62

. True.

(D) Area of

△ P Q_{2} Q_{3} = \frac{1}{2} \cdot 12 \cdot 2 = 62

. False (source says

42

is wrong).

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JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Circle

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