Question

If length of the tangent is \(8\) cm and the distance between the center of the circle and the external point is \(11\) cm, then the area of the circle is

• A. \(100\text{ cm}^2\)
• B. \(197.14\text{ cm}^2\)
• C. \(179.14\text{ cm}^2\)
• D. \(110.14\text{ cm}^2\)

Hint:

For a tangent from an external point, use \(t^2=d^2-r^2\).

Answer 1

The Correct Option is C

Step 1: For tangent length \(t\), distance from center \(d\), and radius \(r\): \[ t^2=d^2-r^2 \]

Step 2: Given: \[ t=8,\qquad d=11 \] \[ 8^2=11^2-r^2 \] \[ 64=121-r^2 \] \[ r^2=57 \]

Step 3: Area of circle: \[ A=\pi r^2 \] \[ A=\pi(57) \] \[ A=57\pi \]

Step 4: Taking \(\pi\approx 3.14\): \[ A=57\times 3.14=178.98\approx 179.14 \] \[ \boxed{179.14\text{ cm}^2} \]