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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Circle

Olympiad workbook · IOQM · Circle

Let _1 be a circle with centre O and let AB be diameter of _1 . Let...

Let

\Omega_1

be a circle with centre O and let AB be diameter of

\Omega_1

. Let P be a point on the segment OB different from O . Suppose another circle

\Omega_2

with centre P lies in the interior of

\Omega_1

. Tangents are drawn from A and B to the circle

\Omega_2

intersecting

\Omega_1

again at

\mathrm{A}_1

and

\mathrm{B}_1

respectively such that

\mathrm{A}_1

and

B_1

are on the opposite sides of

A B

. Given that

A_1 B=5, A B_1=15

and

O P=10

, find the radius of

\Omega_1

.

Let _1 be a circle with centre O and let AB be diameter of _1 . Let P be a point on the segment OB different from O .

Correct answer

20

Step-by-step solution

Let _1 be a circle with centre O and let AB be diameter of _1 . Let P be a point on the segment OB different from O .

\begin{aligned} & \frac{r_1}{r+10}=\frac{5}{2 r} \ldots \ldots \\ & \frac{r_1}{r-10}=\frac{15}{2 r} \ldots \\ & \frac{r-10}{r+10}=\frac{1}{3} \\ & 3 r-30=r+10 \\ & 2 r=40 \\ & r=20 \end{aligned}

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