Question

Let two perpendicular chords of the ellipse  each passing through exactly one of the foci meet at a point P. If from P two tangents are drawn to the hyperbola , then 

Solution

Correct option is

 

The equation of two perpendicular chords drawn through each of the foci be y = m(x – ae) and . Locus of their point of intersection P is obtained by eliminating the variable m.

Multiplying the equations of chords, we have

         y2 = –(x2 – a2e2)   or    x2 + y2 = a2 – b2

Above represents the director circle of hyperbola  which we know is the locus of the point of intersection of perpendicular tangentsQP and QR.

Hence .

SIMILAR QUESTIONS

Q1

Consider a branch of the hyperbola  with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is

Q2

 touches the hyperbola x2 – 2y2 = 4, then the point of contact is

Q3

Equation of tangent to the hyperbola 2x2 – 3y2 = 6 which is parallel to the line y = 3x + 4 is

Q4

Let be two points on the hyperbola . If (h, k) is thepoint of intersection of the normal’s at P and Qk is equal to

Q5

Let be two points on the hyperbola . If (h, k) is thepoint of intersection of the normal’s at P and Qk is equal to

Q6

If x = 9 is the chord of contact of the hyperbola x2 – y2 = 9, then the equation of the corresponding pair of triangle is.

Q7

Find the equations of tangents to the hyperbola x2 – 4y = 36 which are perpendicular to the line x – y + 4 = 0

Q8

Find the coordinates of foci, the eccentricity and latus rectum. Determine also the equation of its directrices for the hyperbola

4x2 – 9y2 =36.

Q9

Find the distance from A(4, 2) to the points in which the line 3x – 5= 2 meets the hyperbola xy = 24. Are these points on the same side of A?

Q10

The asymptotes of the hyperbola  makes an angle 600 with x-axis. Write down the equation of determiner conjugate to the diameter y = 2x.