Home
Class 12
MATHS
Prove that the locus of the point of int...

Prove that the locus of the point of intersection of the tangents at the ends of the normal chords of the hyperbola `x^2-y^2=a^2` is `a^2(y^2-x^2)=4x^2y^2dot`

Text Solution

Verified by Experts

Normal at a point `(a sec theta, a tan theta)` is
`x cos theta+y cot theta=2a`
If `P(x_(1),y_(1))` is the point of intersection of the tangents at the ends of normal chord (1), then (1) must be the chord of contact of P(h, k) whose equation is given by
`hx-ky=a^(2)" (2)"`
Comparing (1) and (2) and eliminating `theta`, we get
`(a^(2))/(4h^(2))-(a^(2))/(4k^(2))=1`
Hence, the locus is
`(1)/(x^(2))-(1)/(y^(2))=(4)/(a^(2))`
Promotional Banner

Topper's Solved these Questions

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.6|4 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise (Single)|68 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.4|5 Videos

  • HIGHT AND DISTANCE

    CENGAGE|Exercise JEE Previous Year|3 Videos

  • INDEFINITE INTEGRATION

    CENGAGE|Exercise Question Bank|25 Videos

Similar Questions

Explore conceptually related problems

The locus of the point of intersection of the tangents at the ends of normal chord of the hyperbola x^(2)-y^(2)=a^(2) is

Prove that the locus of the point of intersection of the tangents at the ends of the normal chords of the hyperbola x^(2)-y^(2)=a^(2) is a^(2)(y^(2)-x^(2))=4x^(2)y^(2)

Locus of the intersection of the tangents at the ends of the normal chords of the parabola y^(2) = 4ax is

The locus of the point of intersection of the tangents at the end-points of normal chords of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 , is

Find the locus of points of intersection of tangents drawn at the end of all normal chords to the parabola y^2 = 8(x-1) .

Find the locus of point of intersection of tangents at the extremities of normal chords of the elipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1

The locus of the point of intersection of tangents drawn at the extremities of normal chords to hyperbola xy=c^(2) is (A)(x^(2)-y^(2))^(2)+4c^(2)xy=0(B)(x^(2)+y^(2))^(2)+4c2^(x)y=0(C)x^(2)-y^(2))^(2)+4c2^(x)y=0(C)x^(2)-y^(2))^(2)+4cxy=0(D)(x^(2)+y^(2))^(2)+4cxy=0

A line through the origin meets the circle x^(2)+y^(2)=a^(2) at P and the hyperbola x^(2)-y^(2)=a^(2) at Q. Prove that the locus of the point of intersection of tangent at P to the circle with the tangent at Q to the hyperbola is a straight line.

The locus of the point of intersection of tangents at the end-points of conjugate diameters of the ellipse x^(2)/a^(2) + y^(2)/b^(2) = 1 , is