Two circles with centres M and N have radii 5 cm and 8 cm, respectively. The circles touch each other externally at point T. A line PR is drawn such that the points M, T and N lie on PR, P being closer to M. From P, a tangent PQ = 12 cm is drawn to the circle with centre M touching at Q, and from R, another tangent RS = 15 cm is drawn to the circle with centre N touching at S. What is the length (in cm) of PR?

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Question

This question was previously asked in

SSC CHSL Exam 2024 Tier-I Official Paper (Held On: 04 Jul, 2024 Shift 3)

Answer 1

Given:

Radius of the circle with center M = 5 cm

Radius of the circle with center N = 8 cm

Tangent PQ = 12 cm

Tangent RS = 15 cm

The circles touch externally at point T.

Formula used:

Use the Pythagorean theorem to find PM and NR:

PM² = PQ² + MQ²

NR² = RS² + NS²

Total length of PR = PM + MN + NR

Calculations:

For PM:

PM² = 12² + 5²

⇒ PM² = 144 + 25

⇒ PM² = 169

⇒ PM = √169 = 13 cm

For NR:

NR² = 15² + 8²

⇒ NR² = 225 + 64

⇒ NR² = 289

⇒ NR = √289 = 17 cm

Now, distance between M and N:

MN = 5 cm + 8 cm = 13 cm

Total length of PR:

PR = PM + MN + NR

⇒ PR = 13 cm + 13 cm + 17 cm = 43 cm

∴ The length of PR is 43 cm.

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