## Question

Find the locus of the mid-pints of the chords of the circle *x*^{2} – *y*^{2} = 16, which are tangent to the hyperbola 9*x*^{2} – 16*y*^{2} = 144.

### Solution

16*x*^{2} – 9*y*^{2}

Let (*n, k*) be the mid- point of the chord of the circle *x*^{2} + *y*^{2} = *a*^{2}.

So that its equation by *T = S*_{1} is *hx + ky = h*^{2} + *k*^{2}

It will touch the hyperbola if c^{2} = *a*^{2}*m*^{2} – *b*^{2}

Replacing a^{2} = 16, b^{2} = 9, *h* by *x* and *k* by *y*, we get :

Locus of mid-point ≡ (*x*^{2} + *y*^{2})^{2} = 16*x*^{2} – 9*y*^{2}

#### SIMILAR QUESTIONS

The equation of normal to the rectangular hyperbola xy = 4 at the point P on the hyperbola which is parallel to the line

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Find the eccentricity of the hyperbola whose latus rectum is half of its transverse axis.

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Determiner the equation of common tangents to the hyperbola and .

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Obtain the equation of a hyperbola with coordinate axes as principal axes given that the distance of one of its vertices from the foci are 9 and 1 units.

Find the equation to the hyperbola of given transverse axes whose vertex bisects the distance between the center and the focus.