Question

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

Solution

Correct option is

(x2 + 4y2)2 = 4(x2 – 4y2)

Any tangent to the given hyperbola is

                                    

Let (h, k) be the middle point of the chord of ellipse.

(i) and (ii) should be identical

 locus is (x2 + 4y2)2 = 4(x2 – 4y2).

SIMILAR QUESTIONS

Q1

If PQ is a double ordinate of the hyperbola  such the OPQ is an equilateral triangle, O being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies –

Q2

A rectangular hyperbola passes through the points A(1, 1), B(1, 5) and C(3, 1). The equation of normal to the hyperbola at A(1, 1) is –

Q3

If a variable line  which is a chord of the hyperbola  subtends a right angle at the centre of the hyperbola then it always touches a fixed circle whose radius is – 

Q4

If values of m for which the line  touches the hyperbola 16x2 – 9y= 144 are the roots of the equation x2 –(a + b)x – 4 = 0, then value of (a + b) is equal to –

Q5

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 –

Q6

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 –

Q7

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 –

   

Q8

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

Q9

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

Q10

Determiner the equation of common tangents to the hyperbola  and .