The locus of the mid-points of the chords of the circle ${ x }^{ 2 }+{ y }^{ 2 }=16$ which are tangent to the hyperbola $9{ x }^{ 2 }-16{ y }^{ 2 }=144$, is
- ${ \left( { x }^{ 2 }+{ y }^{ 2 } \right) }^{ 2 }=16{ x }^{ 2 }+9{ y }^{ 2 }$
- ${ \left( { x }^{ 2 }-{ y }^{ 2 } \right) }^{ 2 }=16{ x }^{ 2 }-9{ y }^{ 2 }$
- ${ \left( { x }^{ 2 }+{ y }^{ 2 } \right) }^{ 2 }=16{ x }^{ 2 }-9{ y }^{ 2 }$
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None of the above
The chord of the circle x^2 + y^2 = 16 with midpoint (h, k) has equation hx + ky = h^2 + k^2. This line is tangent to 9x^2 - 16y^2 = 144 (or x^2/16 - y^2/9 = 1). The condition for tangency y = mx + c to x^2/a^2 - y^2/b^2 = 1 is c^2 = a^2m^2 - b^2. Converting the chord to slope-intercept form and applying the condition leads to the locus (x^2 + y^2)^2 = 16x^2 - 9y^2.
Let the midpoint of a chord of the circle be (h, k); the equation of the chord with this midpoint is hx + ky = h^2 + k^2. The given hyperbola is x^2/a^2 - y^2/b^2 = 1, where a^2 = 16 and b^2 = 9. The condition for the line y = mx + c to be tangent to this hyperbola is c^2 = a^2*m^2 - b^2. Substituting the slope m = -h/k and intercept c = (h^2 + k^2)/k into the tangency condition yields ((h^2 + k^2)/k)^2 = 16*(h^2/k^2) - 9. Multiplying by k^2 and replacing h and k with x and y gives the locus as (x^2 + y^2)^2 = 16x^2 - 9y^2.