Question

If the locus of the mid points of the chords of the circle \(x^2 + y^2 = 25\) that subtend a right angle at the origin is given by \( \frac{x^2}{a^2} + \frac{y^2}{a^2} = 1\), then \(|a| =\)

• A. \( \frac{2}{5} \)
• B. \( \frac{5}{\sqrt{2}} \)
• C. \( \frac{2}{25} \)
• D. \( 5\sqrt{2} \)

Hint:

Remember, for chords subtending a right angle at the circle's center, the perpendicular from the center bisects the chord. This geometrical property simplifies the determination of the locus equation.

Answer 1

The Correct Option is B

Step 1: Consider the equation of the circle \(x^2 + y^2 = 25\). Chords that subtend a right angle at the origin imply that the product of their slopes is \(-1\). 
Step 2: The distance from the origin to any point \((x, y)\) on the circle is 5 (since the radius of the circle is \(\sqrt{25} = 5\)). 
Step 3: A chord subtending a right angle at the origin means the triangle formed by the chord and lines joining its endpoints with the origin is a right-angled isosceles triangle. Thus, the perpendicular from the origin to the chord bisects the chord. 
Step 4: For a chord subtending a right angle at the origin, the midpoint of the chord lies on the locus described by the equation \( \frac{x^2}{a^2} + \frac{y^2}{a^2} = 1 \), where \(a\) is related to the radius of the circle. - Since the radius of the circle is 5, and the locus describes the relationship of the midpoints of chords subtending right angles at the origin, we know that \(a^2\) is half the square of the radius of the circle. This gives: \[ a^2 = \frac{25}{2}. \] Thus, \[ a = \frac{5}{\sqrt{2}}. \]