The circle C1: x2+y2=3, with centre at O, intersects the parabola x2=2 y 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 √3 and centre Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis, then

Question

The circle C1: x2+y2=3, with centre at O, intersects the parabola x2=2 y 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 √3 and centre Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis, then (A) Q 2 Q 3=12 (B) R 2 R 3=4 √6 (C) area of the triangle OR 2 R 3 is 6 √2 (D) area of the triangle PQ 2 Q 3 is 4 √2

Tardigrade Admin · Accepted Answer

Solving the equation of circle and parabola we get point

P (2, 1)

Now equation of tangent at

P (2, 1)

is

2 x + y = 3

Let the centre of the circle

C_{2}

and

C_{3}

are

Q_{2} (0, k_{1})

and

Q_{3} (0, k_{2})

Tangent also touches these circles, so

∣ ∣ \frac{k _{1} - 3}{2 + 1} ∣ ∣ = 23

∴ k_{1} = 9, - 3

So,

Q_{2} (0, 9)

and

Q_{3} (0, - 3)

∴ Q_{2} Q_{3} = 12

Let the point of contact

R_{2}

is

(α_{1}, β_{1})

tangent at

(α_{1}, β_{1})

for circle

C_{2}

α_{1} x + β_{1} y - 9 (y + β_{1}) = - 69

Comparing it with

2 x + y = 3

\frac{α _{1}}{2} = \frac{β _{1} - 9}{1} = \frac{9 β _{1} - 69}{3}

α_{1} = - 22, β_{1} = 7

∴ R_{2} (- 22, 7)

Similarly let

R_{3}

is

(α_{2}, β_{2})

Equation of tangent at

(α_{2}, β_{2})

α_{2} x + β_{2} y + 3 (y + β_{2}) = 3

Comparing with

2 x + y = 3

\frac{α _{2}}{2} = \frac{β _{2} + 3}{1} = \frac{3 - 3 β _{2}}{3}

\Rightarrow α_{2} = 22, β_{2} = - 1

∴ R_{3} (22, - 1)

R_{2} R_{3} = (42)^{2} + 64 = 96 = 46

Area of the triangle

O R_{2} R_{3}

= \frac{1}{2} ∣ ∣ 100 1 - 22 22 17 - 1 ∣ ∣ = 62

Area of the triangle

P Q_{2} Q_{3}

= \frac{1}{2} ∣ ∣ 121109 10 - 3 ∣ ∣ = 62

$Q_{2} Q_{3} = 12$

$R_{2} R_{3} = 46$

area of the triangle $O R_{2} R_{3}$ is $62$

area of the triangle $P Q_{2} Q_{3}$ is $42$

Q2​Q3​=12

R2​R3​=46​

area of the triangle OR2​R3​ is 62​

area of the triangle PQ2​Q3​ is 42​

$Q_{2} Q_{3} = 12$

$R_{2} R_{3} = 46$

area of the triangle $O R_{2} R_{3}$ is $62$

area of the triangle $P Q_{2} Q_{3}$ is $42$