Question

The equation of a circle is \(x^2 + y^2 + 6x - 8y - 24 = 0\). If a chord of the circle subtends an angle of \(60^\circ\) at the centre of the circle, then the length of the chord is

• A. 7 units
• B. 6 units
• C. 5 units
• D. 8 units
• E. 9 units

Hint:

An angle of \(60^\circ\) at the center always implies the chord length is equal to the radius. For an angle of \(90^\circ\), the chord length is \(r\sqrt{2}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The triangle formed by the center of the circle and the endpoints of the chord is an isosceles triangle because two sides are radii. If the angle at the center is \(60^\circ\), the triangle is equilateral.

Step 2: Key Formula or Approach:
1. Radius of circle \(x^2 + y^2 + 2gx + 2fy + c = 0\) is \(r = \sqrt{g^2 + f^2 - c}\).
2. Chord length \(L = 2r \sin(\theta/2)\).

Step 3: Detailed Explanation:
From the circle equation \(x^2 + y^2 + 6x - 8y - 24 = 0\):
\(2g = 6 \implies g = 3\)
\(2f = -8 \implies f = -4\)
\(c = -24\)
Calculate radius \(r\):
\[ r = \sqrt{3^2 + (-4)^2 - (-24)} = \sqrt{9 + 16 + 24} = \sqrt{49} = 7 \]
Given the angle subtended at the center is \(\theta = 60^\circ\).
Since the triangle is equilateral (sides \(r, r, L\) and angle \(60^\circ\)), the chord length \(L\) must equal the radius \(r\).
Alternatively, using the formula:
\[ L = 2 \cdot 7 \cdot \sin(60^\circ/2) = 14 \cdot \sin 30^\circ = 14 \cdot \frac{1}{2} = 7 \]

Step 4: Final Answer:
The length of the chord is 7 units.