Class 10 Maths · Chapter 10

Circles

86 Class 10 Maths questions on Circles from the NCERT textbook, NCERT Exemplar and CBSE board papers 2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 2 of 5

1. Q21 Exemplar · Exercise 9.1
   From a ​point P which is at a ​distance of 13 cm from the centre O of a ​circle of ​radius 5 cm, the pair ​of ​tangents PQ and PR to the circle are drawn. Then the area ​of the quadrilateral ​PQOR is

   1. (a)$60$ cm${}^2$
   2. (b)$65$ cm${}^2$
   3. (c)$30$ cm${}^2$
   4. (d)$32.5$ cm${}^2$
   Show answer

   (a) $60$ cm${}^2$

   Step-by-step solution →
2. Q22 Exemplar · Exercise 9.1
   At ​one ​end A of a diameter AB of a circle of radius 5 cm, tangent XAY is ​drawn to the circle. ​The length of the chord CD parallel ​to XY and at a distance 8 cm from A is

   1. (a)4 cm
   2. (b)5 cm
   3. (c)6 cm
   4. (d)8 cm
   Show answer

   (d) 8 cm

   Step-by-step solution →
3. Q23 Exemplar · Exercise 9.1
   In ​Fig. ​9.5, $AT$ ​is a tangent ​to the circle with ​centre $O$ such that $OT=4\text{cm}$ and $\angle OTA=30^{\circ }$. Then $AT$ ​is equal to

   1. (a)4 cm
   2. (b)2 cm
   3. (c)$2\sqrt{3}$ cm
   4. (d)$4\sqrt{3}$ cm
   Show answer

   (c) $2\sqrt{3}$ cm

   Step-by-step solution →
4. Q24 Exemplar · Exercise 9.1
   In Fig. 9.6, if O is the ​centre of a circle, ​PQ ​is a ​chord and the tangent ​PR ​at P ​makes an angle ​of 50 degrees with PQ, then $\angle POQ$ is equal to

   1. (a)$100^{\circ }$
   2. (b)$80^{\circ }$
   3. (c)$90^{\circ }$
   4. (d)$75^{\circ }$
   Show answer

   (a) $100^{\circ }$

   Step-by-step solution →
5. Q25 Exemplar · Exercise 9.1
   In ​Fig. 9.7, if PA ​and PB are tangents ​to ​the ​circle with centre O such that $\angle APB=50^{\circ }$, ​then $\angle OAB$ is equal ​to

   1. (a)$25^{\circ }$
   2. (b)$30^{\circ }$
   3. (c)$40^{\circ }$
   4. (d)$50^{\circ }$
   Show answer

   (a) $25^{\circ }$

   Step-by-step solution →
6. Q26 Exemplar · Exercise 9.1
   If ​two tangents inclined ​at an angle $60^{\circ }$ are ​drawn to ​a circle of radius 3 cm, then ​length of ​each tangent ​is equal ​to

   1. (a)$\frac{3}{2}\sqrt{3}$ cm
   2. (b)$6$ cm
   3. (c)$3$ cm
   4. (d)$3\sqrt{3}$ cm
   Show answer

   (d) $3\sqrt{3}$ cm

   Step-by-step solution →
7. Q27 Exemplar · Exercise 9.1
   In ​Fig. 9.8, if $PQR$ is the tangent to a ​circle at $Q$ whose centre is $O$, $AB$ ​is ​a chord ​parallel to $PR$ ​and $\angle BQR=70^{\circ }$, ​then $\angle AQB$ is equal to

   1. (a)$20^{\circ }$
   2. (b)$40^{\circ }$
   3. (c)$35^{\circ }$
   4. (d)$45^{\circ }$
   Show answer

   (b) $40^{\circ }$

   Step-by-step solution →
8. Q28 Exemplar · Exercise 9.2
   State whether ​the following is ​true or false ​and give reasons: If ​a chord $AB$ subtends ​an angle of $60^{\circ }$ at ​the centre of a circle, then the angle between the tangents ​at $A$ and $B$ is also $60^{\circ }$.
   Show answer

   False

   Step-by-step solution →
9. Q29 Exemplar · Exercise 9.2
   The length of ​tangent from an external ​point on ​a circle is ​always ​greater than ​the ​radius of the circle.
   Show answer

   False

   Step-by-step solution →
10. Q30 Exemplar · Exercise 9.2
    Write 'True' or 'False' and justify your ​answer: ​The ​length of tangent from an external ​point P on ​a circle with ​centre O ​is always less than ​OP.
    Show answer

    True. In right triangle $OAP$, $O{P}^2=P{A}^2+r^2$ with $r>0$, so $OP>PA$ -- the tangent is always shorter than $OP$.

    Step-by-step solution →
11. Q31 Exemplar · Exercise 9.2
    The angle ​between two tangents to ​a ​circle ​may ​be $0^{\circ }$.
    Show answer

    True

    Step-by-step solution →
12. Q32 Exemplar · Exercise 9.2
    If angle between ​two tangents drawn ​from ​a point $P$ to a ​circle of radius $a$ and ​centre $O$ is $90^{\circ }$, ​then $OP=a\sqrt{2}$. State whether this ​statement is ​true or false and justify.
    Show answer

    True

    Step-by-step solution →
13. Q33 Exemplar · Exercise 9.2
    State whether the following is true or false and ​justify: ​If the angle between two tangents ​drawn from ​a point P ​to ​a circle of radius $a$ and centre ​O ​is $60^{\circ }$, then $OP=a\sqrt{3}$.
    Show answer

    False

    Step-by-step solution →
14. Q34 Exemplar · Exercise 9.2
    The ​tangent ​to ​the circumcircle of an ​isosceles triangle $ABC$ ​at $A$, ​in which $AB=AC$, is parallel ​to $BC$.
    Show answer

    True (proved). The tangent-chord angle equals $\angle ACB$, which equals $\angle ABC$ since $AB=AC$; these equal alternate angles with transversal $AB$ prove the tangent at $A$ is parallel to $BC$.

    Step-by-step solution →
15. Q35 Exemplar · Exercise 9.2
    If ​a number of circles touch a given line ​segment PQ ​at a ​point A, ​then their ​centres lie on the ​perpendicular bisector of PQ.
    Show answer

    False

    Step-by-step solution →
16. Q36 Exemplar · Exercise 9.2
    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 PQ.
    Show answer

    True

    Step-by-step solution →
17. Q37 Exemplar · Exercise 9.2
    State whether the following is true or false and ​justify: ​AB is ​a diameter of ​a circle ​and AC ​is its chord such that $\angle BAC=30^{\circ }$. ​If the tangent at C intersects AB ​extended at D, then $BC=BD$.
    Show answer

    True

    Step-by-step solution →
18. Q38 Exemplar · Exercise 9.3
    Out ​of the two ​concentric ​circles, ​the radius of ​the outer circle is 5 cm and ​the chord $AC$ of length 8 cm ​is a tangent to the inner circle. Find the radius ​of the inner circle.
    Show answer

    The radius of the inner circle is $OM=\sqrt{25-16}=3$ cm.

    Step-by-step solution →
19. Q39 Exemplar · Exercise 9.3
    Two ​tangents PQ and PR ​are drawn from an ​external point to a circle with centre ​O. Prove that ​QORP is ​a ​cyclic quadrilateral.
    Show answer

    $\angle OQP+\angle ORP=90^{\circ }+90^{\circ }=180^{\circ }$, so QORP is a cyclic quadrilateral.

    Step-by-step solution →
20. Q40 Exemplar · Exercise 9.3
    If from an ​external ​point B of a circle with centre O, ​two ​tangents BC and BD ​are drawn such that $\angle DBC=120^{\circ }$, prove that $BC+BD=BO$, i.e., $BO=2BC$.
    Show answer

    $BC+BD=2BC=BO$, proved.

    Step-by-step solution →