Contents

**GSEB Solutions for Class 10 mathematics – Circles (English Medium)**

### Exercise-11.1

**Question 1:**

A and B are the points on ⨀(O, r). \(\overline{AB}\) is not a diameter of the circle. Prove that the tangents to the circle at A and B are not parallel.

**Solution :**

**Question 2:**

A, B are the points on ⨀(O, r) such that tangents at A and B intersect in P. Prove that \(\overrightarrow{OP}\) is the bisector of ∠AOB and \(\overrightarrow{PO}\) is the bisector of ∠APB.

**Solution :**

**Question 3:**

A, B are the points on ⨀(O, r) such that tangents at A and B to the circle intersect in P. Show that the circle with \(\overrightarrow{OP}\) as a diameter passes through A and B.

**Solution :**

**Question 4:**

⨀(O, r_{1}) and ⨀(O, r_{2}) are such that r_{1} > r_{2}. Chord \(\overrightarrow{AB}\) of ⨀(O, r_{1}) touches ⨀(O, r_{2}). Find AB in terms of r_{1} and r_{2} .

**Solution :**

**Question 5:**

In example 4, if r_{1} = 41 and r_{2} = 9, find AB.

**Solution :**

### Exercise-11.2

**Question 1:**

P is the point in the exterior of ⨀(O, r) and the tangents from P to the circle touch the circle at X and Y.

- Find OP, if r = 12, XP = 5
- Find m∠XPO, if m∠XOY = 110
- Find r, if OP = 25 and PY = 24
- Find m∠XOP, if m∠XPO = 80

**Question 1(1):**

**Solution :**

**Question 1(2):**

**Solution :**

**Question 1(3):**

**Solution :**

**Question 1(4):**

**Solution :**

**Question 2:**

Two concentric circles having radii 73 and 55 are given. The chord of the circle with larger radius touches the circle with smaller radius. Find the length of the chord.

**Solution :**

**Question 3:**

\(\overline{AB}\) is a diameter of ⨀(O, 10). A tangent is drawn from B to ⨀(O, 8) which touches ⨀(O, 8) at D. \(\overrightarrow{BD}\) intersects ⨀(O, 10) in C. Find AC.

**Solution :**

**Question 4:**

P is in the exterior of a circle at distance 34 from the centre O. A line through P touches the circle at Q. PQ = 16, find the diameter of the circle.

**Solution :**

**Question 5:**

In figure 11.24, two tangents are drawn to a circle from a point A which is in the exterior of the circle. The points of contact of the tangents are P and Q as shown in the figure. A line l touches the circle at R and intersects \(\overline{AP}\) and \(\overline{AQ}\) in B and C respectively. If AB = c, BC = a, CA = b, then prove that

- AP + AQ = a + b + c
- AB + BR = AC + CR = AP = AQ = \(\frac{a+b+c}{2}\)

**Solution :**

**Question 6:**

Prove that the perpendicular drawn to a tangent to the circle at the point of contact of the tangent passes through the centre of the circle.

**Solution :**

**Question 7:**

Tangents from P, a point in the exterior of ⨀(O, r) touch the circle at A and B. Prove that \(\overline{BD}\bot \overline{AC}\) and \(\overline{OP}\) bisects \(\overline{AB}\).

**Solution :**

**Question 8:**

\(\overleftrightarrow{PT}\) and \(\overleftrightarrow{PR}\) are the tangents drawn to ⨀(O, r) from point P lying in the exterior of the circle and T and R are their points of contact respectively. Prove that m∠TPR = 2m∠OTR.

**Solution :**

**Question 9:**

\(\overline{AB}\) is a chord of ⨀(O, 5) such that AB = 8. Tangents at A and B to the circle intersect in P. Find PA.

**Solution :**

**Question 10:**

P lies in the exterior of ⨀(O, 5) such that OP = 13. Two tangents are drawn to the circle which touch the circle in A and B. Find AB.

**Solution :**

### Exercise-11

**Question 1:**

A circle touches the sides \(\overline{BC}\), \(\overline{CA}\), \(\overline{AB}\) of ∆ABC at points D, E, F respectively. BD = x, CE = y, AF = z. Prove that the area of ∆ABC = \(\sqrt{xyz(x+y+z)} \).

**Solution :**

**Question 2:**

∆ABC is an isosceles triangle in which \(\overline{AB}\) ≅ \(\overline{AC}\). A circle touching all the three sides of ∆ABC touches \(\overline{BC}\) at D. Prove that D is the mid-point of \(\overline{BC}\).

**Solution :**

**Question 3:**

∠B is a right angle in ∆ABC. If AB = 24, BC = 7, then find the radius of the circle which touches all the three sides of ∆ABC.

**Solution :**

**Question 4:**

A circle touches all the three sides of a right angled ∆ABC in which ∠B is right angle. Prove that the radius of the circle is \(\frac{AB+BC+AC}{2}\).

**Solution :**

**Question 5:**

In \(\square \) ABCD, m∠D = 90. A circle with centre O and radius r touches its sides \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\) and \(\overline{DA}\) in P, Q, R and S respectively. If BC = 40, CD = 30 and BP = 25, then find the radius of the circle.

**Solution :**

We know that tangents drawn to a circle are perpendicular to the radius of the circle.

∴ m∠ORD = m∠OSD = 90°

m∠D = 90° (Given)

Also, OR = OS = radius

∴ ⃞ORDS is a square. The tangents drawn to a circle from a point in the exterior of the circle are congruent.

∴ BP = BQ, CQ = CR and DR = DS (by thm.)

Now, BP = BQ

∴ BQ = 25 (∵ BP = 25)

∴ BC – CQ = 25 (∵ BC = 40)

∴ CQ = 15

∴ CR = 15 (∵ CQ=CR)

∴ CD – DR = 15 (∵ C – R – D)

∴ 30 – DR = 15 (∵ given CD = 30)

∴ DR = 15

But ⃞ORDS is a square.

∴ OR = DR = 15

Thus, the radius of a circle is OR=15.

**Question 6:**

Two concentric circles are given. Prove that all chords of the circle with larger radius which touch the circle with smaller radius are congruent.

**Solution :**

**Question 7:**

A circle touches all the sides of \(\square \) ABCD. If AB = 5, BC = 8, CD = 6. Find AD.

**Solution :**

If circle touches all the sides of a quadrilateral, then

AB + CD = BC + DA

∴ 5 + 6 = 8 + DA

∴ 11 = 8 + DA

∴ AD = 3

**Question 8:**

A circle touches all the sides of \(\square \) ABCD. If \(\overline{AB}\) is the largest side then prove that \(\overline{CD}\) is the smallest side.

**Solution :**

**Question 9:**

P is a point in the exterior of a circle having centre O and radius 24. OP = 25. A tangent from P touches the circle at Q. Find PQ.

**Solution :**

**Question 10:**

Select a proper option (a), (b), (c) or (d) from given options :

**Question 10(1):**

P is in exterior of ⨀(0, 15). A tangent from P touches the circle at T. If PT = 8, then OP=……

**Solution :**

a. 17

Here, m∠OTP = 90°

∴ OP^{2 }= OT^{2 }+ TP^{2}

∴ OP^{2 }= (15)^{ 2 }+ (8)^{ 2}

= 255 + 64

= 289

= (17)^{ 2}

∴ OP = 17

**Question 10(2):**

\(\overleftrightarrow{PA}\) , \(\overleftrightarrow{PB}\) touch ⨀(O, r) at A and B. If m∠AOB = 80, then m∠OPB = ……

**Solution :**

**Question 10(3):**

A tangent from P, a point in the exterior of a circle, touches the circle at Q. If OP = 13, PQ = 5, then the diameter of the circle is ……

**Solution :**

d. 24

In right angled ∆OQP.

OP = 13 and PQ = 5.

Now OP^{2 }= OQ^{2 }+ PQ^{2}

∴(13)^{ 2 }= r^{2 }+ (5)^{ 2}

∴r^{2 }= 169 – 25 = 144 = 12^{2}

∴r = 12

Diameter of a circle

= 2× radius

= 2r = 2×12 = 24

**Question 10(4):**

In ∆ABC, AB = 3, BC = 4, AC = 5, then the radius of the circle touching all the three sides is …..

**Solution :**

**Question 10(5):**

\(\overleftrightarrow{PQ}\) and \(\overleftrightarrow{PR}\) touch the circle with centre O at A and B respectively. If m∠OPB = 30 and OP = 10, then radius of the circle = ……..

**Solution :**

**Question 10(6):**

The points of contact of the tangents from an exterior point P to the circle with centre O are A and B. If m∠OPB = 30, then m∠AOB = ………

**Solution :**

d. 120°

In right angled ∆OBP, m∠OPB = 30°.

Further,

m∠BOP + m∠OPB, m∠B = 180° (angles in linear pair)

∴ m∠BOP + 30° + 90° = 180°

∴ m∠BOP = 180° – 120°

∴ m∠BOP = 60°

Now, m∠AOB = 2m∠BOP

= 2×60 = 120°

**Question 10(7):**

A chord of ⨀(O, 5) touches ⨀(O, 3). Therefore the length of the chord = ……….

**Solution :**

a. 8

Here, radius of a smaller circle OM = 3 and the radius of a bigger circle OB = 5.

In right angled ∆OMB,

OB^{2 }= OM^{2 }+ MB^{2}

∴(5)^{ 2 }= (3)^{ 2 }+ MB^{2}

∴MB^{2 }= 25 – 9 = 16

∴MB=4

The length of a chord

AB = 2×MB = 2×4 = 8