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 3 of 5

1. Q41 Exemplar · Exercise 9.3
   Prove ​that the centre of a ​circle touching ​two intersecting ​lines lies on the angle ​bisector ​of the lines.
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   Since $△OAP\cong △OBP$ (RHS), $\angle OPA=\angle OPB$, so $OP$ bisects the angle and the centre $O$ lies on the angle bisector.

   Step-by-step solution →
2. Q42 Exemplar · Exercise 9.3
   In Fig. ​9.13, ​AB and CD ​are common tangents ​to two circles of unequal radii. ​Prove ​that $AB=CD$.

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   $AB=CD$ (proved using equal tangents from the external point $E$).

   Step-by-step solution →
3. Q43 Exemplar · Exercise 9.3
   In Question ​5 above, if radii ​of the ​two circles are ​equal, prove ​that AB $=$ CD.
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   Let $O_1$ and $O_2$ be the centres and $r$ the common radius. A radius is perpendicular to the tangent at the point of contact, so $O_{1}A\perp AB$ and $O_{2}B\perp AB$, with $O_{1}A=O_{2}B=r$. Hence $O_{1}AB{O}_2$ is a rectangle and $AB=O_1{O}_2$. In the same way $O_{1}CD{O}_2$ is a rectangle and $CD=O_1{O}_2$. Therefore $AB=CD$.

   Step-by-step solution →
4. Q44 Exemplar · Exercise 9.3
   In Fig. 9.14, ​common ​tangents $AB$ and $CD$ to ​two circles ​intersect ​at $E$. Prove that $AB=CD$.

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   $AB=CD$ (proved using equal tangents from $E$).

   Step-by-step solution →
5. Q45 Exemplar · Exercise 9.3
   A ​chord ​PQ of a circle is parallel to ​the tangent drawn at a point R of the circle. Prove ​that ​R bisects ​the ​arc PRQ.
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   R bisects the arc PRQ: the tangent-chord angle theorem with the parallel condition gives $\angle RPQ=\angle RQP$, so $RP=RQ$ and therefore arc $RP=$ arc $RQ$.

   Step-by-step solution →
6. Q46 Exemplar · Exercise 9.3
   Prove that the ​tangents drawn at ​the ​ends of ​a ​chord ​of a circle make equal angles with the chord.
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   Since $PA=PB$ (equal tangents from $P$), triangle $PAB$ is isosceles, so $\angle PAB=\angle PBA$ — the tangents make equal angles with the chord.

   Step-by-step solution →
7. Q47 Exemplar · Exercise 9.3
   Prove that ​a diameter AB of a circle bisects all those ​chords which are ​parallel ​to ​the ​tangent ​at ​the point A.
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   Proved: the tangent at A is perpendicular to diameter AB; any chord CD parallel to it is therefore perpendicular to AB, and since AB passes through the centre, it bisects CD.

   Step-by-step solution →
8. Q48 CBSE 2026
   In ​the given ​figure, $PQ$ and $PR$ are tangents ​to a circle ​with centre $O$ ​and radius $3$ cm. ​If $\angle QPR=60^{\circ }$, then the length of each ​tangent is :

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

   Step-by-step solution →
9. Q49 CBSE 2026
   If $PQ$ ​and $PR$ are tangents to the circle with ​centre $O$ and radius $4$ ​cm ​such that $\angle QPR=90^{\circ }$, then the ​length $OP$ ​is ​:

   1. (a)$4$ cm
   2. (b)$4\sqrt{2}$ cm
   3. (c)$8$ cm
   4. (d)$2\sqrt{2}$ cm
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   (b) $4\sqrt{2}$ cm

   Step-by-step solution →
10. Q50 CBSE 2026
    In ​the given ​figure, $PQ$ is tangent ​to the circle with ​centre $O$. $S$ is a ​point on ​the circle such that $\angle SQT=55^{\circ }$. The $m\angle QPS$ is

    1. (a)$55^{\circ }$
    2. (b)$20^{\circ }$
    3. (c)$35^{\circ }$
    4. (d)$70^{\circ }$
    Show answer

    (b) $20^{\circ }$

    Step-by-step solution →
11. Q51 CBSE 2026
    $PT$ ​is ​tangent to the ​circle with ​centre $O$ and radius $5$ cm. $OP$ ​intersects the circle at $Q$. ​If $PQ=x$, then $P{T}^2$ ​equals:

    1. (a)$x^2+5x$
    2. (b)$x^2+10x+50$
    3. (c)$x^2+10x$
    4. (d)$x^2-25$
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    (c) $x^2+10x$

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12. Q52 CBSE 2026
    Four ​tangents drawn to ​a ​circle are ​extended from ​both the sides to form ​a quadrilateral. Which of these quadrilateral is not possible?

    1. (a)Trapezium
    2. (b)Square
    3. (c)Rectangle
    4. (d)Rhombus
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    (c) Rectangle

    Step-by-step solution →
13. Q53 CBSE 2026
    A chord $QR$ ​subtends an angle ​of $105^{\circ }$ ​at the centre $O$ of the ​circle. The measure ​of $\angle RQP$ ​is

    1. (a)$\frac{75^{\circ }}{2}$
    2. (b)$\frac{105^{\circ }}{2}$
    3. (c)$75^{\circ }$
    4. (d)$15^{\circ }$
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    (b) $\frac{105^{\circ }}{2}$

    Step-by-step solution →
14. Q54 CBSE 2026
    If two ​tangents inclined at an ​angle of $60^{\circ }$ are drawn from an external point ​to a ​circle of ​radius $6$ cm, then the length ​of each tangent ​is:

    1. (a)$3\sqrt{3}$ cm
    2. (b)$6$ cm
    3. (c)$12$ cm
    4. (d)$6\sqrt{3}$ cm
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    (d) $6\sqrt{3}$ cm

    Step-by-step solution →
15. Q55 CBSE 2026
    In two concentric ​circles with ​centre $O$, the ​radius of the outer circle is $25$ cm. ​Chord $PQ$ of the ​outer circle is tangent to the ​inner circle at $R$. If $PQ=14$ cm, then the radius of ​the inner ​circle is:

    1. (a)$\sqrt{429}$ cm
    2. (b)$24$ cm
    3. (c)$\sqrt{674}$ cm
    4. (d)$20$ cm
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    (b) $24$ cm

    Step-by-step solution →
16. Q56 CBSE 2026
    If $TP$ and $TQ$ are two ​tangents to ​a ​circle with centre $O$ ​from ​an external point $T$ so ​that $\angle POQ=120^{\circ }$, ​then $\angle PTQ$ is equal to :

    1. (a)$60^{\circ }$
    2. (b)$70^{\circ }$
    3. (c)$80^{\circ }$
    4. (d)$90^{\circ }$
    Show answer

    (a) $60^{\circ }$

    Step-by-step solution →
17. Q57 CBSE 2026
    In ​the given figure, $PT$ is ​a tangent to ​the circle with ​centre $O$ and radius $r$. ​If $\angle POT=45^{\circ }$, ​then the ​length of $OP$ is ​:

    1. (a)$r\sqrt{2}$
    2. (b)$\sqrt{2r}$
    3. (c)$2r$
    4. (d)$r^2$
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    (a) $r\sqrt{2}$

    Step-by-step solution →
18. Q58 CBSE 2026
    From a point $P$, tangents $PQ$ ​and $PR$ ​are ​drawn ​to a circle with centre $O$ and radius $6$ cm. If $OP=10$ ​cm, ​then area of ​quadrilateral $PQOR$ is ​:

    1. (a)$48{\text{cm}}^2$
    2. (b)$24{\text{cm}}^2$
    3. (c)$96{\text{cm}}^2$
    4. (d)$72{\text{cm}}^2$
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    (a) $48{\text{cm}}^2$

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19. Q59 CBSE 2026
    In the given figure, $PQ$ ​and $PR$ are two tangents ​drawn to a circle with centre $O$. If $\angle ORQ=25^{\circ }$, then ​the ​measure of $\angle PQR$ ​is:

    1. (a)$65^{\circ }$
    2. (b)$25^{\circ }$
    3. (c)$50^{\circ }$
    4. (d)$75^{\circ }$
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    (a) $65^{\circ }$

    Step-by-step solution →
20. Q60 CBSE 2026
    In the ​given figure, $PA$ ​is ​a ​tangent ​from an ​external point $P$ to a ​circle with centre $O$. If $\angle POB=125^{\circ }$, then $\angle APO$ ​is equal to :

    1. (a)$25^{\circ }$
    2. (b)$65^{\circ }$
    3. (c)$90^{\circ }$
    4. (d)$35^{\circ }$
    Show answer

    (d) $35^{\circ }$

    Step-by-step solution →