Question

If \( \theta \) is the angle between the pair of tangents drawn from the point \( A(0, 2) \) to the circle \( x^2 + y^2 - 4x + 16y + 88 = 0 \), then \( \tan \theta \) equals:

• A. \( \frac{5}{2} \)
• B. \( 20 \)
• C. \( \frac{4}{5} \)
• D. \( \frac{21}{4} \)

Hint:

For problems involving tangents to a circle, use the formula for the angle between the tangents, ensuring you first complete the square to find the center and radius of the circle.

Answer 1

The Correct Option is B

Step 1: Rewrite the equation of the circle. We are given the equation of the circle: \[ x^2 + y^2 - 4x + 16y + 88 = 0. \] Complete the square for both \( x \) and \( y \): For \( x \): \[ x^2 - 4x = (x - 2)^2 - 4. \] For \( y \): \[ y^2 + 16y = (y + 8)^2 - 64. \] Substitute these into the equation: \[ (x - 2)^2 + (y + 8)^2 = 4. \] Thus, the center of the circle is \( (2, -8) \) and the radius is 2. Step 2: Use the formula for the angle between the tangents. The formula for the angle \( \theta \) between two tangents drawn from an external point \( A(x_1, y_1) \) to a circle with center \( (x_2, y_2) \) and radius \( r \) is: \[ \tan \theta = \frac{r}{\sqrt{d^2 - r^2}}, \] where \( d \) is the distance from the external point to the center of the circle and \( r \) is the radius. Step 3: Calculate the distance from point \( A(0, 2) \) to the center \( (2, -8) \) The distance \( d \) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - 0)^2 + (-8 - 2)^2} = \sqrt{4 + 100} = \sqrt{104}. \] Step 4: Apply the formula for \( \tan \theta \) Now, applying the formula: \[ \tan \theta = \frac{2}{\sqrt{104 - 4}} = \frac{2}{\sqrt{100}} = \frac{2}{10} = \frac{4}{5}. \] Thus, the correct answer is \rupee\rupee(B)\rupee\rupee \( 20 \).