Class 10 Circles Mock Test (CBSE)

Topics in Circles

• Tangent to a circle
• Number of tangents from a point on a circle
• Lengths of tangents from an external point

Circles practice questions with answers

1. Q1Multiple choice·1 mark
   If tangents $PA$ and $PB$ from a point $P$ to a circle with centre $O$ are inclined to each other at an angle of $70^\circ$, then $\angle POA$ is equal to:

   1. (a)$55^\circ$
   2. (b)$50^\circ$
   3. (c)$70^\circ$
   4. (d)$35^\circ$
   Show answer and step marks

   1. Correct option: (a) $55^\circ$. In quadrilateral $PAOB$, $\angle AOB = 180^\circ - 70^\circ = 110^\circ$, and since $OP$ bisects $\angle AOB$, $\angle POA = \frac{1}{2} \times 110^\circ = 55^\circ$.+1

   Answer: (a)
2. Q2Multiple choice·1 mark
   A point T lies inside a circle, not on it. How many tangents can be drawn to the circle from the point T?

   1. (a)One
   2. (b)Two
   3. (c)None
   4. (d)Infinitely many
   Show answer and step marks

   1. Correct option: (c) None+1

   Answer: (c)
3. Q3Assertion and reason·1 mark
   Assertion (A): Tangents drawn at the ends of a diameter of a circle are parallel. Reason (R): The tangent at any point of a circle is perpendicular to the radius through the point of contact.

   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.
   Show answer and step marks

   1. Correct option: (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). Assertion (A) is true because the radii at the ends of a diameter form a straight line, making alternate or interior angles supplementary, thus making tangents parallel. Reason (R) is also true and is the exact property used to prove Assertion (A).+1

   Answer: (a)
4. Q4Assertion and reason·1 mark
   Assertion (A): A point is $5$ cm from the centre of a circle of radius $3$ cm. The length of the tangent from this point to the circle is $4$ cm. Reason (R): The tangent from an external point is always perpendicular to the line joining the point to the centre.

   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.
   Show answer and step marks

   1. Correct option: (c) Assertion (A) is true but Reason (R) is false.+1

   Answer: (c)
5. Q5Short answer·2 marks
   A point $P$ is $25$ cm away from the centre $O$ of a circle of radius $7$ cm. A tangent $PA$ is drawn from $P$ to the circle, touching it at $A$. Find the length of the tangent $PA$.
   Show answer and step marks

   1. The tangent at $A$ is perpendicular to the radius $OA$, so $\angle OAP = 90^\circ$ and $PA^2 = OP^2 - OA^2$+1
   2. $PA^2 = 25^2 - 7^2 = 625 - 49 = 576$+0.5
   3. $PA = \sqrt{576} = 24$ cm+0.5

   Answer: $PA = 24$ cm
6. Q6Short answer·2 marks
   O is the centre of a circle of radius $6$ cm, and P is a point $10$ cm from O. PA and PB are tangents drawn from P, touching the circle at A and B. (a) Write the relation between the lengths of PA and PB. [1] (b) Find the length of PA. [1]
   Show answer and step marks

   1. (a) PA = PB, since the two tangent segments drawn from an external point are equal in length+1
   2. (b) OA is perpendicular to PA, so $PA^2 = OP^2 - OA^2 = 10^2 - 6^2 = 64$, hence $PA = 8$ cm+1

   Answer: (a) $PA = PB$; (b) $PA = 8$ cm

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