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Equation of a Chord Joining Two Points with Circle in Parametric Form

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

C_{1}

C_{2}

externally touch each other and they both internally touch another circle

C_{3}

C_{1}

C_{2}

4

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}

C_{2}

. Find the length of the chord

C_{1}

C_{2}

are externally tangent and they are both internally tangent to the circle

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C_{1}

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}

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)

C_{1}

C_{2}

are externally tangent and they are both internally tangent to the circle

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C_{1}

C_{2}

4

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}

C_{2} .

Given that the length of the chord is

\frac{m \sqrt{n}}{p}

m, n

p

are positive integers,

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p

are relatively prime and

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is not divisible by the square of any prime, find the value of

(m + n + p) .

Q. Three circles

C_{1}, C_{2}

C_{3}

r_{1}, r_{2}

r_{1} + r_{2}

respectively are such that

C_{1}

C_{2}

touch each other externally and

C_{3}

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r_{3}

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r_{1} > r_{2} > r_{3}

r_{1}, r_{2}, r_{3}

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{(\frac{r_{1}}{r_{2}})}^{3}

Q. The centers of two circles

C_{1} a n d C_{2}

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C_{1}

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C_{2}

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