From an external point P, the tangent PA is drawn to the circle with centre O.

(i) If OP = 20 cm and tangent PA = 16 cm find the diameter of the circle.

(ii) If diameter of the circle is 20 cm and tangent PA = 24 cm, find the length of OP.

O is center of the circle. PA is the tangent of the circle.

We know that,

Tangent and radius through the center at point of contact are perpendicular to each other.

(i) Given, OP = 20 cm and PA = 16 cm

In the ∆OPA as we know that ∠OAP = 90°.

As we know Pythagoras theorem can be used in a right angled triangle,

⇒ OA² + PA² = OP²

⇒ OA² + 16² = 20²

⇒ OA² + 256 = 400

⇒ OA² = 400 - 256

⇒ OA² = 144

⇒ OA = $\sqrt{144}$

⇒ OA = 12 cm

By formula,

Diameter = 2 x radius = 2 x 12 = 24 cm.

Hence, the diameter of the circle = 24 cm.

(ii) Given, diameter = 20 cm and PA = 24 cm

Radius (OA) = $\dfrac{1}{2} \times 20$ = 10 cm

In the ∆ OPA as we know that ∠OAP = 90°.

As we know Pythagoras theorem can be used in a right angled triangle,

⇒ OA² + PA² = OP²

⇒ 10² + 24² = OP²

⇒ 100 + 576 = OP²

⇒ OP² = 676

⇒ OP = $\sqrt{676}$

⇒ OP = 26 cm.

Hence, the length of OP = 26 cm.