Question

A tangent line L is drawn at the point \( (2, -4) \) on the parabola \( y^2 = 8x \). If the line L is also tangent to the circle \( x^2 + y^2 = a \), then 'a' is equal to _________.

Hint:

For a line \( lx + my + n = 0 \) to be tangent to \( x^2 + y^2 = a \), the condition is \( a = \frac{n^2}{l^2 + m^2} \).

Answer 1

Correct Answer: 2

Step 1: Understanding the Concept:
We first find the equation of the tangent to the parabola at a specific point. For this line to also be a tangent to a circle, the perpendicular distance from the center of the circle to the line must equal the radius of the circle.
Step 2: Detailed Explanation:
Equation of tangent to \( y^2 = 8x \) at \( (x_1, y_1) = (2, -4) \): \[ yy_1 = 4(x + x_1) \implies -4y = 4(x + 2) \implies y = -x - 2 \] Line L: \( x + y + 2 = 0 \).
This line is tangent to the circle \( x^2 + y^2 = a \). Center of circle = \( (0, 0) \), radius \( r = \sqrt{a} \). Distance from origin to line L: \[ r = \frac{|0 + 0 + 2|}{\sqrt{1^2 + 1^2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \] Since \( r = \sqrt{a} \), we have \( \sqrt{a} = \sqrt{2} \implies a = 2 \).
Step 3: Final Answer:
The value of a is 2.