Chapter 9 · Circles

NCERT Exemplar Class 10 Maths Exercise 9.2

All 10 questions of NCERT Exemplar Class 10 Maths Exercise 9.2 (Circles), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   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 →
2. Q2
   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 →
3. Q3
   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 →
4. Q4
   The angle ​between two tangents to ​a ​circle ​may ​be $0^{\circ }$.
   Show answer

   True

   Step-by-step solution →
5. Q5
   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 →
6. Q6
   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 →
7. Q7
   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 →
8. Q8
   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 →
9. Q9
   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 →
10. Q10
    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 →