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, r1) and ⨀(O, r2) are such that r1 > r2. Chord \(\overrightarrow{AB}\) of ⨀(O, r1) touches ⨀(O, r2). Find AB in terms of r1 and r2 .
Solution :
Question 5:
In example 4, if r1 = 41 and r2 = 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°
∴ OP2 = OT2 + TP2
∴ OP2 = (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 OP2 = OQ2 + PQ2
∴(13) 2 = r2 + (5) 2
∴r2 = 169 – 25 = 144 = 122
∴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,
OB2 = OM2 + MB2
∴(5) 2 = (3) 2 + MB2
∴MB2 = 25 – 9 = 16
∴MB=4
The length of a chord
AB = 2×MB = 2×4 = 8