Circles — Problems & Practice

Chapter 10 | CBSE Class 10 Mathematics (NCERT-style practice)

Original practice inspired by NCERT ideas (not verbatim exercises).

Easy

E1

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

Solution
Exactly one.

E2

A tangent from $Q$ has length $24$ cm and $O Q = 25$ cm. Find the radius.

Solution $r = 2 5^{2} - 2 4^{2} = 625 - 576 = 49 = 7 cm .$

E3

Fill in: A tangent intersects a circle in ____ point(s); a secant intersects in ____ point(s).

Solution
One; two.

E4

Prove that the tangents at the ends of a diameter are parallel.

Solution
Let diameter be $A B$ , centre $O$ , tangents at $A$ and $B$ .
By Theorem 10.1, tangent at $A$ $⊥ O A$ and tangent at $B$ $⊥ O B$ .
But $O A$ and $O B$ are the same straight line, so both tangents are perpendicular to the same line $\Rightarrow$ they are parallel.

Medium

M1

Two tangents $T P$ and $T Q$ are drawn to a circle with centre $O$ . If $∠ P O Q = 11 0^{\circ}$ , find $∠ P T Q$ .

Solution
In quadrilateral $O P T Q$ : $∠ O P T = ∠ O QT = 9 0^{\circ}$ , $∠ P O Q = 11 0^{\circ}$ .
$∠ P T Q = 36 0^{\circ} - 9 0^{\circ} - 9 0^{\circ} - 11 0^{\circ} = 7 0^{\circ} .$ (Equivalently: $∠ P T Q = 18 0^{\circ} - ∠ P O Q$ .)

M2

Length of a tangent from $A$ is $4$ cm and $O A = 5$ cm. Find the radius.

Solution $r = 5^{2} - 4^{2} = 3 cm .$

M3

Two concentric circles have radii $5$ cm and $3$ cm. Find the length of a chord of the larger circle that touches the smaller circle.

Solution
Let chord $A B$ touch smaller circle at $P$ . Then $O P ⊥ A B$ and $P$ mid-point of $A B$ .
$A P = 5^{2} - 3^{2} = 4 \Rightarrow A B = 8 cm .$

M4

From external point $P$ , tangents $P A$ and $P B$ touch a circle at $A$ and $B$ . If $∠ A P B = 8 0^{\circ}$ , find $∠ P O A$ (centre $O$ ).

Solution
$△ O A P$ is right-angled at $A$ ; also $O P$ bisects $∠ A P B$ , so $∠ A P O = 4 0^{\circ}$ .
$∠ P O A = 9 0^{\circ} - 4 0^{\circ} = 5 0^{\circ} .$

Hard

H1

A quadrilateral $A B C D$ circumscribes a circle. Prove $A B + C D = A D + B C$ .

Solution
Let points of contact on $A B, B C, C D, D A$ give equal tangent segments from each vertex:
From $A$ : both tangents $= a$ ; from $B$ : $= b$ ; from $C$ : $= c$ ; from $D$ : $= d$ .
Then $A B + C D = (a + b) + (c + d)$ and $A D + B C = (a + d) + (b + c)$ — equal.

H2

Prove that the parallelogram circumscribing a circle is a rhombus.

Solution
From H1, $A B + C D = A D + B C$ . In parallelogram $A B = C D$ , $A D = B C$ , so $2 A B = 2 A D$ $\Rightarrow$ $A B = A D$ . All sides equal $\Rightarrow$ rhombus.

H3

Chord $P Q = 8$ cm in a circle of radius $5$ cm; tangents at $P$ and $Q$ meet at $T$ . Find $T P$ .

Solution
Let $O T$ meet $P Q$ at $R$ . Then $R$ mid-point, $P R = 4$ , $O R = 5^{2} - 4^{2} = 3$ .
Right $△ T R P \sim △ P R O$ (AA): $\frac{T P}{5} = \frac{4}{3} \Rightarrow T P = \frac{20}{3} cm .$

Self-check

State Theorems 10.1 and 10.2 without notes.
Sketch the $0/1/2$ tangents cases.
Derive tangent length $O P^{2} - r^{2}$ .
Re-prove H1 using equal tangent segments.
Why are tangents at diameter ends parallel?