There are two circles which touch each other externally. The radius of the first circle with centre O is 17 cm and radius of the second circle with centre A is 7 cm. BC is a direct common tangent to these two circles, where B and C are points on the circles with centres O and A, respectively. The length of BC is:

Question

Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 08 Dec 2022 Shift 4)

Answer 1

Given:

Radius of first circle = 17 cm

Radius of second circle = 7 cm

Circles touch each other externally.

BC is a direct common tangent of these two circles.

Concept used:

External tangents are lines that do not cross the segment joining the centers of the circles.

Calculation:

F1 Savita SSC 19-5-23 D2

Join A to O and B. Join O to C. Draw AQ perpendicular to OC.

Now, AQ = BC, as they are opposite sides of rectangle AQCB.

⇒AO = AP + PQ

⇒AO = 7 + 17

⇒AO = 24 cm

⇒OQ = CO - CQ

⇒OQ = 17 - 7

⇒OQ = 10 cm

Now, applying pythagoras theorem in triangle AOQ:

⇒\(AO^2\) = \(AQ^2+OQ^2\)

⇒AQ = \(\sqrt{AO^2-OQ^2}\)

⇒AQ = \(\sqrt{24^2-10^2}\)

⇒AQ = \(2\sqrt{119}\)

∴ The length of BC is \(2\sqrt{119}\).

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