R.D. Sharma solutions for मैथमैटिक्स [अंग्रेजी] कक्षा १० chapter 8 - Circles [Latest edition]

The common point of a tangent to a circle and the circle is called the ______.

A circle may have ______ parallel tangents.

A tangent to a circle intersects it in ______ point (s).

A line intersecting a circle in two points is called a ______.

The angle between tangent at a point on a circle and the radius through the point is ______.

How many tangents can a circle have?

O is the center of a circle of radius 8cm. The tangent at a point A on the circle cuts a line through O at B such that AB = 15 cm. Find OB

If the tangent at point P to the circle with center O cuts a line through O at Q such that PQ = 24 cm and OQ = 25 cm. Find the radius of circle.

If PT is a tangent at T to a circle whose center is O and OP = 17 cm, OT = 8 cm. Find the length of tangent segment PT.

A point P is 26 cm away from O of circle and the length PT of the tangent drawn from P to the circle is 10 cm. Find the radius of the circle.

In the given figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q. If PA = 12 cm, QC = QD = 3 cm, then find PC + PD.

If AB, AC, PQ are tangents in Fig. and AB = 5 cm find the perimeter of ΔAPQ.

In Fig below, PQ is tangent at point R of the circle with center O. If ∠TRQ = 30°. Find ∠PRS.

In figure, a circle touches all the four sides of a quadrilateral ABCD whose sides AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD.

In figure, there are two concentric circles with centre O of radii 5 cm and 3 cm. From an external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find the length of BP.

In the figure, a circle is inscribed in a quadrilateral ABCD in which ∠B = 90°. If AD = 23 cm, AB = 29 cm and DS = 5 cm, find the radius r of the circle.

A circle with centre O and radius 8 cm is inscribed in a quadrilateral ABCD in which P, Q, R, S are the points of contact as shown. If AD is perpendicular to DC, BC = 30 cm and BS = 24 cm, then find the length DC.

In the given figure, AB is a chord of length 16 cm of a circle of radius 10 cm. The tangents at A and B intersect at a point P. Find the length of PA.

Two concentric circles are of diameters 30 cm and 18 cm. Find the length of the chord of the larger circle which touches the smaller circle.

If from any point on the common chord of two intersecting circles, tangents be drawn to circles, prove that they are equal.

Two circles touch externally at a point P. From a point T on the tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. Prove that TQ = TR.

In figure, PA and PB are tangents from an external point P to the circle with centre O. LN touches the circle at M. Prove that PL + LM = PN + MN.

In the given figure, ABC is a right triangle right-angled at B such that BC = 6 cm and AB = 8 cm. Find the radius of its incircle.

In the given figure, BDC is a tangent to the given circle at point D such that BD = 30 cm and CD = 7 cm. The other tangents BE and CF are drawn respectively from B and C to the circle and meet when produced at A making BAC a right angle triangle. Calculate (i) AF 

In the given figure, BDC is a tangent to the given circle at point D such that BD = 30 cm and CD = 7 cm. The other tangents BE and CF are drawn respectively from B and C to the circle and meet when produced at A making BAC a right angle triangle. Calculate (ii) radius of the circle.

In figure, the tangent at a point C of a circle and a diameter AB when extended intersect at P. If ∠PCA = 110°, find ∠CBA and ∠BCO.

[Hint: Join CO.]

In Fig. 8.79, PQ is a tangent from an external point P to a circle with centre O and OP cuts the circle at T and QOR is a diameter. If ∠POR = 130° and S is a point on the circle, find ∠1 + ∠2.

In the given figure, a ∆ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 8 cm and 6 cm respectively. Find the lengths of sides AB and AC, when area of ∆ABC is 84 cm². 

In the given figure, AB is a diameter of a circle with centre O and AT is a tangent. If \[\angle\] AOQ = 58º, find  \[\angle\] ATQ.

In figure, OQ : PQ = 3 : 4 and perimeter of ΔPDQ = 60 cm. Determine PQ, QR and OP.

In Fig. 8.78, there are two concentric circles with centre O. PRT and PQS are tangents to the inner circle from a point P lying on the outer circle. If PR = 5 cm, find the length of PS.

If PA and PB are tangents from an outside point P. such that PA = 10 cm and ∠APB = 60°. Find the length of chord AB.

From an external point P , tangents PA = PB are drawn to a circle with centre O   . If  \[\angle PAB = {50}^o\] , then find  \[\angle AOB\]

The lengths of three consecutive sides of a quadrilateral circumscribing a circle are 4 cm, 5 cm and 7 cm respectively. Determine the length of the fourth side.

In the given figure, O is the centre of the circle and BCD is tangent to it at C. Prove that ∠BAC + ∠ACD = 90°.

In the following figure, PB is a tangent to the circle with centre O at B. AB is a chord of the circle of length 24 cm and at a distance of 5 cm from the centre of the circle. If the length PB of the tangent is 20 cm, find the length of OP.

Rectangle ABCD circumscribes the circle of radius 10 cm. Prove that ABCD is a square. Hence, find the perimeter of ABCD.

In the adjoining figure, TP and TQ are tangents drawn to a circle with centre O. If ∠OPQ = 15° and ∠PTQ = θ, then find the value of sin 2θ.

Equal circles with centres O and O' touch each other at X. OO' produced to meet a circle with centre O', at A. AC is a tangent to the circle whose centre is O. O'D is perpendicular to AC. Find the value of\[\frac{DO'}{CO}\]

In the given figure, BC is a tangent to the circle with centre O. OE bisects AP. Prove that ΔAEO ∼ Δ ABC. 

In the given figure, PO \[\perp\]  QO. The tangents to the circle at P and Q intersect at a point T. Prove that PQ and OTare right bisector of each other.

In the following figure, O is the centre of the circle and BCD is tangent to it at C. Prove that ∠BAC + ∠ACD = 90°.

From a point P two tangents PA and PB are drawn to a circle with centre at O. If OP = 2r, show that ΔPAB is equilateral.

A triangle PQR is drawn to circumscribe a circle of radius 8 cm such that the segments QT and TR, into which QR is divided by the point of contact T, are of lengths 14 cm and 16 cm respectively. If area of ∆PQR is 336 cm², find the sides PQ and PR.

In the following figure, PA and PB are tangents to the circle drawn from an external point P. CD is a third tangent touching the circle at Q. If PB = 10 cm and CQ = 2 cm, what is the length PC?

Two tangents TP and TQ are drawn from an external point T to a circle with centre O as shown in figure. If they are inclined to each other at an angle of 100°, then what is the value of ∠POQ?

In the following figure, CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the circle at R. If CP = 11 cm and BC = 7 cm, then find the length of BR.

The length of tangent from a point A at a distance of 5 cm from the centre of the circle is 4 cm. What is the radius of the circle?

What the distance between two parallel tangents to a circle of radius 5 cm?

In Q. No. 1, if PB = 10 cm, what is the perimeter of ΔPCD?

What is the distance between two parallel tangents of a circle of radius 4 cm?

In the following figure, ΔABC is circumscribing a circle. Find the length of BC.

In the following figure, CP and CQ are tangents from an external point C to a circle with centre O. AB is another tangent which touches the circle at R. If CP = 11 cm and BR = 4 cm, find the length of BC.

Hint: Given, CP = 11 cm.

∴ CP = CQ ⇒ CQ = 11cm.

Tangents drawn point B are equal.

∴ BR = BQ ⇒ BQ = 4 cm.

Hence, BC = CQ – BQ = (11 – 4) cm = 7 cm.

Two concentric circles are of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle.

In figure, PA and PB are tangents to the circle with centre O such that ∠APB = 50°. Write the measure of ∠OAB.

In the following, PQ is a chord of a circle and PT is the tangent at P such that ∠QPT = 60°. Then, find ∠PRQ.

In the following figure, PQL and PRM are tangents to the circle with centre O at the points Q and R respectively and S is a point on the circle such that ∠SQL = 50° and ∠SRM = 60°. Then, find ∠QSR.

In the following figure, BOA is a diameter of a circle and the tangent at a point P meets BA produced at T. If ∠PBO = 30°, then find ∠PTA. Also, show that AP = AT.

In the following figure, AB is diameter of a circle centered at O. BC is tangent to the circle at B. If OP bisects the chord AD and ∠AOP = 60°, then find m∠C.

In the following figure, XAY is a tangent to the circle centered at O. If ∠ABO = 40°, then find m∠BAY  and m∠AOY.

In the following figure, if tangents PA and PB drawn from a point P to a circle with centre O, are inclined to each other at an angle of 70°, then find the measure of ∠POA.

AB and AC are tangents drawn from a point A to a circle with centre O. If ∠BAC = 65°, then find the measure of ∠BOC.

In the given figure, PA is a tangent to the circle drawn from the external point P and PBC is the secant to the circle with BC as diameter. If ∠AOC = 130°, then find the measure of ∠APB, where O is the centre of the circle.

In the given figure, AB and CD are tangents to a circle centred at O. Is ∠BAC = ∠DCA? Justify your answer. 

If two tangents inclined at angle of 60° are drawn to a circle of radius 3 cm, then find the length of each tangents.

The length of a chord of a bigger circle concentric with a circle of radius 3 cm is 8 cm. If the chord is a tangent to the smaller circle, then find the radius of the bigger circle.

In the following, ΔPQR circumscribes the circle. If PX = 10 cm, QY = 8 cm and RZ = 6 cm, then find the perimeter of ΔPQR.

A person is standing at P outside a circular ground at a distance of 26 m from the centre of the ground. He found that his distances from the points A and B on the ground are 10 m (PA and PB are tangents to the circle). Find the radius of the circular ground.

At point A on the diameter AB of a circle of radius 10 cm, tangent XAY is drawn to the circle. Find the length of the chord CD parallel to XY at a distance of 16 cm from A.

A line intersecting a circle in two distinct points is called a ______.

The tangents drawn, at the end-points of a diameter, to a circle are ______.

The lengths of two tangents drawn from an external point to a circle are ______.

The two tangents drawn from an external point to a circle are ______ to the segment joining the centre to the point.

Tangents drawn from an external point to a circle subtend ______ angles to the centre.

Parallelogram circumscribing a circle is a ______.

If AB and CD are common tangents to two circles of unequal radii, then ______.

From an external point P, two tangents PA and PB are drawn to the circle with centre O. Then the angle between OP and AB is ______.

If angle between two tangents drawn from a point P to a circle of radius a and centre O is 90°, then OP = ______.

AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = ______.

If a chord AB subtends an angle of 60° at the centre of a circle, then the angle between the tangents at A and B is ______.

The length of the tangent from an external point P on the circle with centre O is always ______ OP.

If the angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = ______.

If a number of circles pass through the end points P and Q of a line segment PQ, then their centres lie on the perpendicular bisector of ______.

The angle between two tangents drawn from an external point to a circle is ______ to the angle subtended by the line segments joining the points of contact at the centre.

PA and PB are two tangents drawn from an external point P to a circle with centre O. If ∠PBA = 65°, then ∠APB = ______.

PA and PB are two tangents drawn from an external point P to a circle with centre O. If ∠APB = 80° then ∠POA = ______.

If PA and PB are two tangents drawn from an external point P to a circle such that PA = 5 cm and ∠APB = 60°, then the length of chord AB is ______.

Quadrilateral ABCD is circumscribed to a circle. If AB = 6 cm, BC = 7 cm and CD = 4 cm, then AD = ______.

The length of the tangent from an external point P to a circle of radius 5 cm is 10 cm. The distance of the point from the centre of the circle is ______.