The line 2 x - y intersects the circle C 1- x 2- y 2-5 at P in the 1st quadrant- C 2 and C 3 are circles- both with radii equal to 2 -5 with respective centers Q 2 and Q 3 lying on the y-axis- If the tangent to C 1 at P touches the circles C 2 and C 3 then the distance Q 2 Q 3 is

Point of intersection of the curves nbsp;C_1:x^2+y^2=5 and nbsp;2x=y nbsp; is (1,2) Equation of the tangent at (1,2) ⇒ x+2y=5 Let Q_2 and Q_3 be (0,k) ⇒|1 ...

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Byju's Answer

Standard XII

Mathematics

Focal Chord

The line 2 x ...

Question

The line $2 x = y$ intersects the circle $C_{1} : x^{2} + y^{2} = 5$ at P in the 1st quadrant. $C_{2}$ and $C_{3}$ are circles, both with radii equal to $2 \sqrt{5}$ with respective centers $Q_{2}$ and $Q_{3}$ lying on the y-axis. If the tangent to $C_{1}$ at $P$ touches the circles $C_{2}$ and $C_{3}$ then the distance $Q_{2} Q_{3}$ is

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Solution

Point of intersection of the curves $C_{1} : x^{2} + y^{2} = 5$ and $2 x = y$ is $(1, 2)$
Equation of the tangent at $(1, 2)$
$\Rightarrow x + 2 y = 5$
Let $Q_{2}$ and $Q_{3}$ be $(0, k)$
$\Rightarrow ∣ ∣ ∣ \frac{1 \times 0 + 2 \times k - 5}{\sqrt{5}} ∣ ∣ ∣ = 2 \sqrt{5}$
$\Rightarrow 2 k - 5 = \pm 10 \Rightarrow k = 7.5, - 2.5 \Rightarrow Q_{2} = (0, 7.5), Q_{3} = (0, - 2.5)$
$\Rightarrow$ Distance $Q_{2} Q_{3} = 10 units$

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