Chapter 10 : Circles
Q. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
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Q. Prove that the parallelogram circumscribing a circle is a rhombus.
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Q. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
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Q. 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 at the centre.
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Q. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is-
1. 7 cm
2. 12 cm
3. 15 cm
4. 24.5 cm
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Q.
Fill in the Blanks:
(i) A tangent to a circle intersects it in ________ point(s).
(ii) A line intersecting a circle in two points is called a _______.
(iii) A circle can have _______ parallel tangents at the most.
(iv) A common point of a tangent to a circle and the circle is called ______.
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Q. Why did it make Mr Purcell feel “vaguely insulted”?
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Q. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
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Q. If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80o, then POA is equal to.
1. 50o
2. 60o
3. 80o
4. 70o
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Q. In Figure, XY and XY are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and XY at B. Prove that AOB=90o.

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Q. How many tangents can a circle have?
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Q. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of the lengths 8 cm and 6 cm respectively. Find the sides AB and AC.

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Q. A tangent PQ at a point P of a circle of radius 5cm meets a line through the centre O at a point Q so that OQ=12 cm. Length PQ is:
1. 12 cm
2. 8.5 cm
3. 13 cm
4. 119 cm
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Q. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
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Q. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
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Q. Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
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Q. In Figure If TP and TQ are the two tangents to a circle with centre O so that POQ =110o, the PTQ is equal to.

1. 70o
2. 90o
3. 80o
4. 60o
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