## Question

The locus of the mid-point of the chord of the circle *x*^{2} + *y*^{2 }= 16, which are tangent to the hyperbola 9*x*^{2} – 16*y*^{2 }= 144 is

### Solution

(*x*^{2} + *y*^{2})^{2 }= 16*x*^{2} – 9*y*^{2}

9*x*^{2} – 16*y*^{2 }= 144 ⇒ Equation of tangent to hyperbola is

…… (1)

…… (2)

Let (*x*_{1}, *y*_{1}) be mid-point of the chord of circle *x*^{2} + *y*^{2 }= 16 equation of this is

*x x*_{1} + *y y*_{1} *–* (*x*_{1}^{2} + *y*_{1}^{2}) = 0 …… (3)

so

eliminate *m* to get locus of (*x*_{1}, *y*_{1}) as

(*x*^{2} + *y*^{2})^{2 }= 16*x*^{2} – 9*y*^{2}

#### SIMILAR QUESTIONS

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.

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