Question

In the given figure, TP and TQ are tangents to a circle with centre M, touching another circle with centre N at A and B respectively. It is given that MQ = 13 cm, NB = 8 cm, BQ = 35 cm and TP = 80 cm.

1. Name the quadrilateral MQBN.   [1]
2. Is MN parallel to PA? Justify your answer.   [1]
3. Find length TB.   [1]
4. Find length MN.   [2]

Answer 1

i. In quadrilateral MQBN, MQ ⊥ TQ and NB ⊥ TQ (Radius ⊥ Tangent).

Since both are perpendicular to the same line TQ, MQ || NB.

A quadrilateral with one pair of opposite sides parallel is a Trapezium.

ii. No, MN is not parallel to PA. PA is a chord/segment on the tangents, while MN is the line joining the centers. There is no geometric condition satisfyng parallelism here.

iii. Since TP and TQ are tangents from T to the circle with center M, TP = TQ = 80 cm.

Now, TQ = TB + BQ.

80 = TB + 35

⇒ TB = 45 cm

iv. We know MQ || NB. To find the distance between centers MN in trapezium MQBN:

Draw a line from N perpendicular to MQ, say at point X.

QX = NB = 8 cm.

MX = MQ – QX

= 13 – 8

= 5 cm

In right △MXN, NX = BQ = 35 cm.

MN² = MX² + NX²

= 5² + 35²

= 25 + 1225

= 1250

MN = `sqrt(1250)`

= `25sqrt(2)`

= 35.36 cm