Question

Find the locus of the mid-pints of the chords of the circle x2 – y2 = 16, which are tangent to the hyperbola 9x2 – 16y2 = 144.

Solution

Correct option is

16x2 – 9y2

Let (n, k) be the mid- point of the chord of the circle x2 + y2 = a2.

So that its equation by T = S1 is hx + ky = h2 + k2

            

It will touch the hyperbola if c2 = a2m2 – b2

Replacing a2 = 16, b2 = 9, h by x and k by y, we get :

Locus of mid-point ≡ (x2 + y2)2 = 16x2 – 9y2

SIMILAR QUESTIONS

Q1

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

2x – y = 5 is –

Q2

A tangent to the hyperbola  meets ellipse x2 + 4y2 = 4 in two distinct points. Then the locus of midpoint of this chord is –

Q3

From a point on the line y = x + c, c(parameter), tangents are drawn to the hyperbola  such that chords of contact pass through a fixed point (x1, y1). Then  is equal to –

Q4

If the portion of the asymptotes between centre and the tangent at the vertex of hyperbola  in the third quadrant is cut by the line  being parameter, then –

   

Q5

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

Q6

For what value of c does not line y = 2x + c touches the hyperbola 16x2 – 9y2 = 144?

Q7

Determiner the equation of common tangents to the hyperbola  and .

Q8

Find the locus of the poles of the normal of the hyperbola .

Q9

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.

Q10

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