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Standard XII

Mathematics

Length of Common Chord

Circle C 1 ...

Question

Circle $C_{1}$ and $C_{2}$ are externally tangent and they are both internally tangent to the circle $C_{3}$ . The radii of $C_{1}$ and $C_{2}$ are 4 and 10, respectively and the centres of the three circles are collinear. A chord of $C_{3}$ is also a common internal tangent of $C_{1}$ and $C_{2}$ . Given that the length of the chord is $\frac{m \sqrt{n}}{p}$ where m, n and p are positive integers, such that $\frac{m \sqrt{n}}{p}$ is in its simplest form, find the value of $(m + n + p)$ .

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Solution

The possible diagram of the given problem is shown
From the diagram the radius of bigger circle will be $10 + 4 = 14$
We have to find the length of $D E$
$D E = 2 \times M D$
Triangle $C M D$ is right angled triangle. (since tangent is perpendicular to radius at point of contact)
$C M = A M - A C$
$C M = 10 - 4 = 6$
${C M}^{2} + {M D}^{2} = {C D}^{2}$
$6^{2} + {M D}^{2} = 14^{2}$
${M D}^{2} = 160 M D = 4 \sqrt{10}$
$E D = 8 \sqrt{10}$
Comparing with $\frac{m \sqrt{n}}{p}$ , $m = 8, n = 10, p = 1$
So our answer will be $8 + 10 + 1 = 19$ .

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