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

To find the locus of the midpoints of the chords of the circle \(x^2 + y^2 = 16\) that are tangent to the hyperbola \(9x^2 - 16y^2 = 144\), we follow these steps:

The equation of the circle is \(x^2 + y^2 = 16\). The general form for a chord of the circle, whose midpoint is \((h, k)\), is given by the equation:

1. \(T = S_1\)

The equation of the chord with midpoint \((h,k)\) is:

1. \(xx_1 + yy_1 = 16\),

where \((x_1, y_1)\) correspond to the endpoints of the chord on the circle.

For this chord to be tangent to the hyperbola \(9x^2 - 16y^2 = 144\), the chord should satisfy the condition:

1. \(\frac{x^2_1}{16} - \frac{y^2_1}{9} = 1\)

The locus of the point \((h, k)\)—the midpoint—can be found by substituting \(x = h + x_1\) and \(y = k + y_1\) ensuring that it satisfies both the circle and hyperbola's conditions.

By solving the equations of the chord and the hyperbola together using substitution and eliminating parameters, we find:

1. \({(x^2 + y^2)^2} = 16x^2 - 9y^2\)

Therefore, the correct option that represents the locus of the midpoints of these chords is:

1. \((x^2 + y^2)^2 = 16x^2 - 9y^2\)

Thus, the answer is: (x^2 + y^2)^2 = 16x^2 - 9y^2.