Question

If the pair of tangents drawn to the circle \( x^2 + y^2 = a^2 \) from the point \( (10, 4) \) are perpendicular, then \( a \) is:

• A. \( \sqrt{58} \)
• B. \( 58 \)
• C. \( 2\sqrt{63} \)
• D. \( 2\sqrt{45} \)

Hint:

For perpendicular tangents, use the property \( h^2 + k^2 = 2a^2 \).

Answer 1

The Correct Option is A

Step 1: Understanding the Condition for Perpendicular Tangents The given equation of the circle is: \[ x^2 + y^2 = a^2 \] A pair of tangents drawn from an external point \( (x_1, y_1) \) to a circle are perpendicular if and only if the given point lies on the director circle of the given circle.
Step 2: Equation of the Director Circle The equation of the director circle of a given circle \( x^2 + y^2 = a^2 \) is given by: \[ x^2 + y^2 = 2a^2 \] Since the point \( (10,4) \) lies on the director circle, we substitute \( x = 10 \) and \( y = 4 \): \[ 10^2 + 4^2 = 2a^2 \] \[ 100 + 16 = 2a^2 \]
Step 3: Solving for \( a \) Dividing by 2: \[ a^2 = 58 \] \[ a = \sqrt{58} \]
Final Answer: \[ \boxed{\sqrt{58}} \]