Question

If two tangents inclined at an angle of \(60^\circ\) are drawn from an external point to a circle of radius 6 cm, then the length of each tangent is :

• A. \(3\sqrt{3}\) cm
• B. 6 cm
• C. 12 cm
• D. \(6\sqrt{3}\) cm

Hint:

In circle problems involving tangents from an external point:
If the inclination between tangents is \(60^\circ\), the triangle formed by the two tangents and the chord of contact is equilateral.
Using \(\tan(30^\circ) = \frac{\text{Radius}}{\text{Tangent}}\) directly gives:
\[ \text{Tangent} = \text{Radius} \times \sqrt{3} \]
which makes the calculation virtually instant!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a circle of radius 6 cm. From an external point \(P\), two tangents \(PA\) and \(PB\) are drawn to the circle. The angle between the two tangents is \(60^\circ\). We need to determine the length of each tangent.

Step 2: Key Formula or Approach:
1. Tangents from an external point to a circle are equal in length, so \(PA = PB\).
2. The tangent at any point on a circle is perpendicular to the radius through the point of contact, meaning \(\angle OAP = 90^\circ\).
3. The line joining the center of the circle to the external point bisects the angle between the two tangents.
4. We can use basic trigonometry in the right-angled triangle \(OAP\) to find the length of the tangent.

Step 3: Detailed Explanation:
1. Let \(O\) be the center of the circle. Let \(A\) and \(B\) be the points of contact of the tangents from the external point \(P\).
2. Given that radius \(OA = 6\) cm and the angle between the tangents \(\angle APB = 60^\circ\).
3. The line segment \(OP\) bisects \(\angle APB\):
\[ \angle APO = \frac{\angle APB}{2} = \frac{60^\circ}{2} = 30^\circ \]
4. In right-angled triangle \(OAP\) (right-angled at \(A\)):
\[ \tan(\angle APO) = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{OA}{PA} \]
5. Substitute the values:
\[ \tan(30^\circ) = \frac{6}{PA} \]
6. Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{6}{PA} \]
\[ PA = 6\sqrt{3} \text{ cm} \]
Since both tangents are equal, the length of each tangent is \(6\sqrt{3}\) cm.

Step 4: Final Answer:
The length of each tangent is \(6\sqrt{3}\) cm, which corresponds to option (D).