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Length of Common Chord

Circles C1 ...

Question

Circles $C_{1}$ & $C_{2}$ externally touch each other and they both internally touch another circle $C_{3}$ . The radii of $C_{1}$ & $C_{2}$ are $4$ and $10$ respectiveley, and the centers of the three circles are collinear. A chord of $C_{3}$ is a transverse common tangent to the circles $C_{1}$ and $C_{2}$ . Find the length of the chord

A

4 \sqrt{10}

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B

14

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C

8 \sqrt{5}

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D

None of these

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Solution

The correct option is D None of these
$O_{3} P = r_{3} = \frac{10 + 10 + 4 + 4}{2} = 14 O_{3} R = 14 - 4 - 4 = 6 P R = \sqrt{14^{2} - 6^{2}} = 4 \sqrt{10}$
$P Q = 8 \sqrt{10}$ units.

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Similar questions

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Let $c_{1} : x^{2} + y^{2} = 1; C_{2} : (x - 10)^{2} + y^{2} = 1 a n d C_{3}; x^{2} + y^{2} - 10 x - 42 y + 457 = 0$ be three circle.A circle C has been drawn to touch circles $C_{1}$ and $C_{2}$ externally and $C_{3}$ internally. Now circles $C_{1}, C_{2}$ and $C_{3}$ start rolling on the circumference of circle C in anticlockwise direction with constant speed. The centroid of the triangle formed by joining the centres of rolling circles $C_{1}, C_{2}$ and $C_{3}$ lies on

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