Circles: 1 mark Questions (CBSE Class 10)

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)$x^2-25$

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)$75^{\circ }$

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)$15^{\circ }$

PQ is tangent to a circle at a point P on the circle. The number of tangents which can be drawn to the circle parallel to PQ, is

1. (A)2
2. (B)1
3. (C)many
4. (D)zero

PQ is tangent to the circle with centre O such that OP = 2OQ. $m\angle OPQ$ is

1. (A)$15^{\circ }$
2. (B)$60^{\circ }$
3. (C)$45^{\circ }$
4. (D)$30^{\circ }$

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)$90^{\circ }$

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)$35^{\circ }$

In the given figure, PA and PB are tangents to a circle centred at O. If $\angle OAB=15^{\circ }$, then $\angle APB$ equals :

1. (A)$30^{\circ }$
2. (B)$15^{\circ }$
3. (C)$45^{\circ }$
4. (D)$10^{\circ }$

In the given figure, PA and PB are tangents to a circle centred at O. If $\angle AOB=130^{\circ }$, then $\angle APB$ is equal to :

1. (A)$130^{\circ }$
2. (B)$50^{\circ }$
3. (C)$120^{\circ }$
4. (D)$90^{\circ }$

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

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)$r^2$

Assertion (A) : Radius is the smallest distance of a tangent from the centre of the circle.
Reason (R) : Radius is perpendicular to the tangent.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

PA and PB are tangents to a circle centred at O. If $\angle PBA=65^{\circ }$, then $\angle APB$ equals :

1. (A)$65^{\circ }$
2. (B)$60^{\circ }$
3. (C)$50^{\circ }$
4. (D)$35^{\circ }$

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

PQ is tangent to a circle with centre O. If $\angle POR=65^{\circ }$, then $m\angle PTR$ is

1. (A)$65^{\circ }$
2. (B)$58.5^{\circ }$
3. (C)$57.5^{\circ }$
4. (D)$45^{\circ }$

From an external point P, a tangent PT has been drawn to a circle with centre at O and radius 3 cm, intersecting its concentric circle at A and B. If AB = 8 cm and OA = AP, the length PQ equals.

1. (A)8 cm
2. (B)10 cm
3. (C)9 cm
4. (D)12 cm

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)$70^{\circ }$

PQ is tangent to a circle with centre O. If OQ = a, OP = a + 2 and PQ = 2b, then relation between a and b is

1. (A)$a^2+(a+2{)}^2=(2b{)}^2$
2. (B)$b^2=a+4$
3. (C)$2a^2+1=b^2$
4. (D)$b^2=a+1$

In the given figure, PA is a tangent to a circle with centre O. If OP = 10 cm, then the length of AP is :

1. (A)$10\sqrt{3}$ cm
2. (B)20 cm
3. (C)5 cm
4. (D)$5\sqrt{3}$ cm

If the distance of a tangent to a circle from its centre is 4 cm, then the length of diameter of the circle is :

1. (A)2 cm
2. (B)4 cm
3. (C)8 cm
4. (D)16 cm

In the given figure, if AB is a tangent to the circle with centre O such that OB = 6 cm and $\angle AOB=60^{\circ }$, then the length of OA is :

1. (A)3 cm
2. (B)$3\sqrt{3}$ cm
3. (C)$4\sqrt{3}$ cm
4. (D)12 cm

In two concentric circles, a tangent to the smaller circle will intersect the larger circle at :

1. (A)zero point
2. (B)one point
3. (C)two points
4. (D)three points

In the given figure, from which point can the tangent not be drawn to the circle with radius $r$ ?

1. (A)X
2. (B)P
3. (C)Q
4. (D)Y

In the given figure, chord AB of the larger circle touches the smaller circle at C. If both the circles have the same centre O, then the length of BD is :

1. (A)1 cm
2. (B)2 cm
3. (C)3 cm
4. (D)4 cm

How many tangents can be drawn from the point P on the outer circle to the inner circle in the given figure ?

1. (A)0
2. (B)1
3. (C)2
4. (D)Infinite

PQ and PR are tangents to the circle of radius 3 cm and centre O. If length of each tangent is 4 cm, then perimeter of $△$ OQP is :

1. (A)5 cm
2. (B)12 cm
3. (C)9 cm
4. (D)8 cm

PA and PB are tangents to a circle with centre O. If $\angle AOB=105^{\circ }$ then $\angle OAP+\angle APB$ is equal to :

1. (A)$75^{\circ }$
2. (B)$175^{\circ }$
3. (C)$180^{\circ }$
4. (D)$165^{\circ }$

In the given figure, AB is diameter of the circle with centre O. CD is tangent to the circle so that $\angle ACD=40^{\circ }$. The value of $\angle CBA$ is

1. (A)$50^{\circ }$
2. (B)$40^{\circ }$
3. (C)$80^{\circ }$
4. (D)$45^{\circ }$

PQ and PR are tangents to a circle with centre O such that OQ = QP. The value of $\angle OPQ$ is equal to

1. (A)$45^{\circ }$
2. (B)$30^{\circ }$
3. (C)$60^{\circ }$
4. (D)$90^{\circ }$

PQ is tangent to the circle centred at O. If OQ = 3 cm, PQ = 5 cm, then OP is equal to

1. (A)5 cm
2. (B)4 cm
3. (C)$\sqrt{15}$ cm
4. (D)$\sqrt{34}$ cm

Tangent BC makes an angle of $x^{\circ }$ with the chord AB of circle centered at O. If $\angle AOB=100^{\circ }$, then value of $x$ is

1. (A)40
2. (B)80
3. (C)90
4. (D)50

In the given figure, PA is tangent to a circle with centre O. If $\angle APO=30^{\circ }$ and OA = 2.5 cm, then OP is equal to

1. (A)2.5 cm
2. (B)5 cm
3. (C)$\frac{5}{\sqrt{3}}$ cm
4. (D)2 cm

In two concentric circles centred at O, a chord AB of the larger circle touches the smaller circle at C. If OA = 3.5 cm, OC = 2.1 cm, then AB is equal to

1. (A)5.6 cm
2. (B)2.8 cm
3. (C)3.5 cm
4. (D)4.2 cm

The tangents drawn at the extremities of the diameter of a circle are always :

1. (A)parallel
2. (B)perpendicular
3. (C)equal
4. (D)intersecting

In the given figure, $PA$ is a tangent from an external point $P$ to a circle with centre $O$. If $\angle POB=115^{\circ }$, then $\angle APO$ is equal to :

1. (A)$25^{\circ }$
2. (B)$65^{\circ }$
3. (C)$90^{\circ }$
4. (D)$35^{\circ }$

If the length of a chord of a circle is equal to its radius, then the angle subtended by chord at the centre is :

1. (A)$60^{\circ }$
2. (B)$30^{\circ }$
3. (C)$120^{\circ }$
4. (D)$90^{\circ }$

In the given figure, RS is the tangent to the circle at the point L and MN is the diameter. If $\angle NML=30^{\circ }$, then $\angle RLM$ is :

1. (A)$30^{\circ }$
2. (B)$60^{\circ }$
3. (C)$90^{\circ }$
4. (D)$120^{\circ }$

If tangents PA and PB drawn from an external point P to the circle with centre O are inclined to each other at an angle of $80^{\circ }$ as shown in the given figure, then the measure of $\angle POA$ is :

1. (A)$40^{\circ }$
2. (B)$50^{\circ }$
3. (C)$60^{\circ }$
4. (D)$80^{\circ }$

Assertion (A) : If two tangents are drawn to a circle from an external point, then they subtend equal angles at the centre of the circle.
Reason (R): A parallelogram circumscribing a circle is a rhombus.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

In the adjoining figure, AB is the chord of the larger circle touching the smaller circle. The centre of both the circles is O. If $AB=2r$ and $OP=r$, then the radius of larger circle is :

1. (A)$2r$
2. (B)$3r$
3. (C)$2\sqrt{2}r$
4. (D)$\sqrt{2}r$

A parallelogram having one of its sides 5 cm circumscribes a circle. The perimeter of parallelogram is :

1. (A)20 cm
2. (B)less than 20 cm
3. (C)more than 20 cm but less than 40 cm
4. (D)40 cm

Assertion (A) : Tangents drawn at the end points of a diameter of a circle are always parallel to each other.
Reason (R) : The lengths of tangents drawn to a circle from a point outside the circle are always equal.

1. (A)Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
2. (B)Both, Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. (C)Assertion (A) is true but Reason (R) is false.
4. (D)Assertion (A) is false but Reason (R) is true.

In the adjoining figure, AB is the chord of larger circle which touches the smaller circle at P. If length of $AB=$ diameter of inner circle $=2r$, then the diameter of larger circle is :

1. (A)$2r$
2. (B)$4r$
3. (C)$2\sqrt{2}r$
4. (D)$\sqrt{2}r$

Assertion (A) : A line drawn perpendicular to the tangent at point of contact passes through the centre of the circle.
Reason (R) : Lengths of tangents drawn from external point to a circle are equal.

1. (A)Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
2. (B)Both, Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. (C)Assertion (A) is true but Reason (R) is false.
4. (D)Assertion (A) is false but Reason (R) is true.

In the adjoining figure, AC is diameter of larger circle with centre O. AB is tangent to smaller circle with centre O. If $OD=r$, then BC is equal to :

1. (A)$r$
2. (B)$\frac{3r}{2}$
3. (C)$2r$
4. (D)$4r$

Which of the following statements is false ?

1. (A)Infinite number of tangents can be drawn to a circle.
2. (B)Infinite number of tangents can be drawn to a circle from a point outside the circle.
3. (C)Infinite number of secants can be drawn to a circle from a point outside the circle.
4. (D)Angle between tangent and diameter at point of contact is $90^{\circ }$.

In the adjoining figure, PA and PB are tangents to a circle with centre O. The measure of angle APB is

1. (A)$210^{\circ }$
2. (B)$150^{\circ }$
3. (C)$105^{\circ }$
4. (D)$30^{\circ }$

In the adjoining figure, the sum of radii of two concentric circles is 16 cm. The length of chord AB which touches the inner circle at P is 16 cm. The difference of the radii of the given circles is

1. (A)8 cm
2. (B)4 cm
3. (C)2 cm
4. (D)3 cm

In the adjoining figure, PA and PB are tangents to a circle with centre O such that $\angle P=90^{\circ }$. If $AB=3\sqrt{2}$ cm, then the diameter of the circle is

1. (A)$3\sqrt{2}$ cm
2. (B)$6\sqrt{2}$ cm
3. (C)3 cm
4. (D)6 cm

For a circle with centre O and radius 5 cm, which of the following statements is true ?
P : Distance between every pair of parallel tangents is 5 cm.
Q : Distance between every pair of parallel tangents is 10 cm.
R : Distance between every pair of parallel tangents must be between 5 cm and 10 cm.
S : There does not exist a point outside the circle from where length of tangent is 5 cm.

1. (A)P
2. (B)Q
3. (C)R
4. (D)S

In the adjoining figure, TS is a tangent to a circle with centre O. The value of $2x^{\circ }$ is

1. (A)22.5
2. (B)45
3. (C)67.5
4. (D)90

In the adjoining figure, AP and AQ are tangents to the circle with centre O. If reflex $\angle POQ=210^{\circ }$, the value of $2x$ is

1. (A)$30^{\circ }$
2. (B)$60^{\circ }$
3. (C)$120^{\circ }$
4. (D)$300^{\circ }$

Assertion (A) : If PA and PB are tangents drawn to a circle with centre O from an external point P, then the quadrilateral OAPB is a cyclic quadrilateral.
Reason (R) : In cyclic quadrilateral opposite angles are equal.

1. (A)Both, Assertion (A) and Reason (R) are true. Reason (R) explains Assertion (A) completely.
2. (B)Both, Assertion (A) and Reason (R) are true. Reason (R) does not explain Assertion (A).
3. (C)Assertion (A) is true but Reason (R) is false.
4. (D)Assertion (A) is false but Reason (R) is true.

Assertion (A) : If the PA and PB are tangents drawn to a circle with centre O from an external point P, then the quadrilateral OAPB is a cyclic quadrilateral.
Reason (R) : In a cyclic quadrilateral, opposite angles are equal.

1. (A)Both, Assertion (A) and Reason (R) are true. Reason (R) explains Assertion (A) completely.
2. (B)Both, Assertion (A) and Reason (R) are true. Reason (R) does not explain Assertion (A).
3. (C)Assertion (A) is true but Reason (R) is false.
4. (D)Assertion (A) is false but Reason (R) is true.

In the given figure, tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of $80^{\circ }$. $\angle ABO$ is equal to

1. (A)$40^{\circ }$
2. (B)$80^{\circ }$
3. (C)$100^{\circ }$
4. (D)$50^{\circ }$

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

1. (A)secant
2. (B)chord
3. (C)diameter
4. (D)tangent

In the given figure, tangents PA and PB drawn from P to circle are inclined to each other at an angle of $80^{\circ }$. The measure of $\angle PAB$ is

1. (A)$80^{\circ }$
2. (B)$60^{\circ }$
3. (C)$50^{\circ }$
4. (D)$40^{\circ }$

From an external point P, a tangent PA is drawn to a circle. The number of tangents through P parallel to PA is :

1. (A)2
2. (B)more than 2
3. (C)1
4. (D)0

In the given figure, PA is a tangent from an external point P to a circle with centre O. If $\angle AOP=70^{\circ }$, then the measure of $\angle APO$ is :

1. (A)$70^{\circ }$
2. (B)$90^{\circ }$
3. (C)$110^{\circ }$
4. (D)$20^{\circ }$

A tangent to a circle is a line that touches the circle at :

1. (A)one point only
2. (B)two points
3. (C)three points
4. (D)infinite number of points

If the angle between the two radii of a circle is $130^{\circ }$, then the angle between the tangents at the ends of these radii, is :

1. (A)$50^{\circ }$
2. (B)$60^{\circ }$
3. (C)$90^{\circ }$
4. (D)$130^{\circ }$

AP and AQ are tangents drawn from an external point A to a circle with centre O and inclined to each other at an angle of $90^{\circ }$. If the length of each tangent is 2 cm, then the radius of the circle is :

1. (A)4 cm
2. (B)2 cm
3. (C)$2\sqrt{2}$ cm
4. (D)1 cm

Assertion (A) : The tangents drawn at the end points of a diameter of a circle, are parallel.
Reason (R) : Diameter of a circle is the longest chord.

1. (A)Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
2. (B)Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A).
3. (C)Assertion (A) is true but Reason (R) is false.
4. (D)Assertion (A) is false but Reason (R) is true.

In the given figure, tangents PA and PB to the circle centred at O, from point P are perpendicular to each other. If PA = 5 cm, then length of AB is equal to

1. (A)5 cm
2. (B)$5\sqrt{2}$ cm
3. (C)$2\sqrt{5}$ cm
4. (D)10 cm

In the given figure, AT is tangent to a circle centred at O. If $\angle CAT=40^{\circ }$, then $\angle CBA$ is equal to

1. (A)$70^{\circ }$
2. (B)$50^{\circ }$
3. (C)$65^{\circ }$
4. (D)$40^{\circ }$

In the given figure, PT is tangent to a circle with centre O. Chord PQ subtends an angle of $65^{\circ }$ at the centre. The measure of $\angle QPT$ is :

1. (A)$65^{\circ }$
2. (B)$57.5^{\circ }$
3. (C)$67.5^{\circ }$
4. (D)$32.5^{\circ }$

Maximum number of common tangents that can be drawn to two circles intersecting at two distinct points is :

1. (A)4
2. (B)3
3. (C)2
4. (D)1

In the given figure, if PT is a tangent to a circle with centre O and $\angle TPO=35^{\circ }$, then the measure of $\angle x$ is :

1. (A)$110^{\circ }$
2. (B)$115^{\circ }$
3. (C)$120^{\circ }$
4. (D)$125^{\circ }$

In the given figure, O is the centre of the circle. MN is the chord and the tangent ML at point M makes an angle of $70^{\circ }$ with MN. The measure of $\angle MON$ is :

1. (A)$120^{\circ }$
2. (B)$140^{\circ }$
3. (C)$70^{\circ }$
4. (D)$90^{\circ }$

In the given figure, AB and AC are tangents to the circle. If $\angle ABC=42^{\circ }$, then the measure of $\angle BAC$ is :

1. (A)$96^{\circ }$
2. (B)$42^{\circ }$
3. (C)$106^{\circ }$
4. (D)$86^{\circ }$

In the given figure, QR is a common tangent to the two given circles touching externally at A. The tangent at A meets QR at P. If AP = 4.2 cm, then the length of QR is :

1. (A)4.2 cm
2. (B)2.1 cm
3. (C)8.4 cm
4. (D)6.3 cm

A chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is :

1. (A)$5\sqrt{2}$
2. (B)$10\sqrt{2}$
3. (C)$\frac{5}{\sqrt{2}}$
4. (D)5

In the given figure, RJ and RL are two tangents to the circle. If $\angle RJL=42^{\circ }$, then the measure of $\angle JOL$ is :

1. (A)$42^{\circ }$
2. (B)$84^{\circ }$
3. (C)$96^{\circ }$
4. (D)$138^{\circ }$

The distance between two parallel tangents of a circle of diameter 7 cm is :

1. (A)7 cm
2. (B)14 cm
3. (C)$\frac{7}{2}$ cm
4. (D)28 cm

In the given figure, the perimeter of $△ABC$ is :

1. (A)30 cm
2. (B)15 cm
3. (C)45 cm
4. (D)60 cm

In the given figure, BC and BD are tangents to the circle with centre O and radius 9 cm. If OB = 15 cm, then the length (BC + BD) is :

1. (A)18 cm
2. (B)12 cm
3. (C)24 cm
4. (D)36 cm

Assertion (A) : A tangent to a circle is perpendicular to the radius through the point of contact.
Reason (R) : The lengths of tangents drawn from the external point to a circle are equal.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

From a point P, two tangents PQ and PR are drawn to a circle with centre at O. T is a point on the major arc QR of the circle. If $\angle QPR=50^{\circ }$, then $\angle QTR$ equals :

1. (A)$50^{\circ }$
2. (B)$130^{\circ }$
3. (C)$65^{\circ }$
4. (D)$90^{\circ }$

How many tangents can be drawn to a circle from a point on it ?

1. (A)One
2. (B)Two
3. (C)Infinite
4. (D)Zero

The length of the tangent from an external point A to a circle, of radius 3 cm, is 4 cm. The distance of A from the centre of the circle is :

1. (A)7 cm
2. (B)5 cm
3. (C)$\sqrt{7}$ cm
4. (D)25 cm

Assertion (A) : A tangent to a circle is perpendicular to the radius through the point of contact.
Reason (R) : The lengths of tangents drawn from an external point to a circle are equal.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) gives the correct explanation of Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true but Reason (R) does not give the correct explanation of Assertion (A).
3. (C)Assertion (A) is true but Reason (R) is false.
4. (D)Assertion (A) is false but Reason (R) is true.

A circle of radius 5.2 cm has two tangents AB and CD parallel to each other. What is the distance between the two tangents ?

1. (A)5.2 cm
2. (B)10.4 cm
3. (C)20.8 cm
4. (D)can’t find

In the given figure, O is the centre of a circle, PQ is a chord and the tangent PT at P makes an angle of $50^{\circ }$ with PQ. The measure of $\angle POQ$ is :

1. (A)$130^{\circ }$
2. (B)$100^{\circ }$
3. (C)$90^{\circ }$
4. (D)$75^{\circ }$