Circles Questions

 Circles 

1. The tangent at a point C of a circle and a diameter AB when extended intersect at P. 
If `angle`PCA=110º , find `angle`CBA [see Fig. ].

2. In Fig. O is the centre of a circle of radius 5 cm, T is a point such that OT = 13 cm and OT intersects the circle at E. If AB is the tangent to the circle at E, find the length of AB.

3. In Fig. common tangents AB and CD to two circles intersect at E. Prove that AB = CD.

4. In Fig, tangents `PQ` and `PR` are drawn to a circle such that `angleRPQ = 30°`. A chord `RS` is drawn parallel to the tangent `PQ.` Find the `angleRQS.`

Ans: 

Given 

PR and PQ are the tangents drawn from P. 

∠RPQ = 30°

Hence, PR = PQ (Tangents drawn from an external point are equal)

 ∴ `anglePRQ = anglePQR`

In `trianglePQR`

`anglePRQ + anglePQR + angleRPQ = 180°`

2`anglePQR = 150°`

`therefore anglePQR = 75°`

`anglePQR = angleRSQ` (Alternate Segment Theorem)

Given `RS∥ PQ`

`therefore anglePQR = angleSRQ = 75° ` (Alternate Interior Angles)

⇒`triangle QRS` is an isosceles triangle.

In `triangleQRS`

`angleSRQ + angleRSQ + angleRQS = 180°`

`75° + 75° + angleRQS = 180°`

`angleRQS = 180° - 150°`

`thereforeRQS = 30°`

5. From a point `P`, two tangents `PA` and `PB` are drawn to a circle `C(0, r)`. If `OP = 2r`, then find `angleAPB`. What type of triangle is` APB`?

6. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact to the centre.

7. Two tangents `TP` and `TQ` are drawn to a circle with centre `O` from an external point `T`. Prove that `anglePTQ= 2angleOPQ`.