How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

Learn MoreLet \( \Omega_1 \) be a circle with center O and let AB be a diameter of \( \Omega_1 \). Let P be a point on the segment OB different from O. Suppose another circle \( \Omega_2 \) with center 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 \(A_1\) and \(B_1\) respectively such that \(A_1 \) and \(B_1\) are on the opposite sides of AB. Given that \(A_1B = 5, AB_1 = 15 \) and \( OP = 10\), find the radius of \( \Omega_1 \).

Do you really need a hint? Try it first!

Hint 1

Draw a diagram carefully.

Hint 2

Suppose the point of tangencies are at C and D. Join PC and PD.

Can you find **two pairs of similar triangles?**

Hint 3

\( \Delta APC \sim \Delta AA_1B \)

Why?

Notice that AC is perpendicular to \( AA_1 \) as the radius is perpendicular to the tangent.

Also \( \angle A \) is common to both triangles. Hence the two triangles are similar (equiangular implies similar).

Similarly \( \Delta BPD \sim \Delta BAB_1 \).

Use the ratio of sides to find OA.

Hint 4

Suppose OA = R (radius of the big circle).

OC = r (radius of the small circle).

We already know OP = 10, \( A_1 B = 5, AB_1 = 15\)

Since \( \Delta AA_1B \) and \( ACP \) are similar we have \( \frac{AP}{AB} = \frac{PC}{A_1B}\). This implies \( \frac{R+10}{2R} = \frac{r}{5}\) **(1)**

Similarlly since \( \Delta BPD \) and \( BAB_1 \) are similar we have \( \frac{BP}{BA} = \frac{PD}{AB_1}\). This implies \( \frac{R-10}{2R} = \frac{r}{15}\) **(2)**

Multiply the reciprocal of (2) with (1) to get R = 20.

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year.

Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

JOIN TRIAL