Question

The locus of the mid-points of the chords of the circle \(x^2 + y^2 = 16\) which are tangent to the hyperbola \(9x^2 - 16y^2 = 144\) is

• A. \((x^2 + y^2)^2 = 16x^2 - 9y^2\)
• B. \((x^2 - y^2)^2 = 16x^2 - 9y^2\)
• C. \((x^2 + y^2)^2 = 16x^2 + 9y^2\)
• D. None of the above

Hint:

Equation of chord with midpoint \((h, k)\) for circle \(x^2 + y^2 = a^2\) is \(xh + yk = h^2 + k^2\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Chord of circle with given midpoint \((h, k)\) has equation \(T = S_1\). This chord is tangent to hyperbola.

Step 2: Detailed Explanation:
Circle: \(x^2 + y^2 = 16\). Midpoint \((h, k)\). Equation of chord: \(xh + yk = h^2 + k^2\).
Hyperbola: \(9x^2 - 16y^2 = 144\). Condition for tangency: \(c^2 = a^2 m^2 - b^2\) for slope form. Convert chord to slope form: \(y = -\frac{h}{k}x + \frac{h^2 + k^2}{k}\).
Slope \(m = -\frac{h}{k}\), intercept \(c = \frac{h^2 + k^2}{k}\).
Tangency condition for hyperbola \(\frac{x^2}{16} - \frac{y^2}{9} = 1\): \(c^2 = a^2 m^2 - b^2 = 16m^2 - 9\).
Substitute: \(\left(\frac{h^2 + k^2}{k}\right)^2 = 16\left(\frac{h^2}{k^2}\right) - 9\)
\(\frac{(h^2 + k^2)^2}{k^2} = \frac{16h^2 - 9k^2}{k^2}\)
\((h^2 + k^2)^2 = 16h^2 - 9k^2\).
Locus: \((x^2 + y^2)^2 = 16x^2 - 9y^2\).

Step 3: Final Answer:
Option (A).