Home
Class 12
MATHS
A point P moves such that the chord of c...

A point `P` moves such that the chord of contact of the pair of tangents from `P` on the parabola `y^2=4a x` touches the rectangular hyperbola `x^2-y^2=c^2dot` Show that the locus of `P` is the ellipse `(x^2)/(c^2)+(y^2)/((2a)^2)=1.`

Text Solution

Verified by Experts

The equation of the chord of contact of parabola w.r.t. `P(x_(1),y_(1))` is given by
`yy_(1)=2a(x+x_(1))" (1)"`
Equation (1) touches the curve
`x^(2)-y^(2)=c^(2)" (2)"`
Using the condition of tangency on
`y=(2ax)/(y_(1))+(2ax_(1))/(y_(1))`
we get
`(4a^(2)x_(1)^(2))/(y_(1))=c^(2)(4a^(2))/(y_(1)^(2))-c^(2)" "[because"(1) is tangent to"(x^(2))/(c^(2))-(y^(2))/(c^(2))=1]`
`"or "4a^(2)x_(1)^(2)=4a^(2)c^(2)-c^(2)y_(1)^(2)`
`"or "4a^(2)x_(1)^(2)+c^(2)y_(1)^(2)=4a^(2)c^(2)`
Hence, the locus is
`(x^(2))/(c^(2))+(y^(2))/((2a)^(2))=1`
which is an ellipse.
Promotional Banner

Topper's Solved these Questions

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.4|5 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.5|5 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.2|12 Videos

  • HIGHT AND DISTANCE

    CENGAGE|Exercise JEE Previous Year|3 Videos

  • INDEFINITE INTEGRATION

    CENGAGE|Exercise Question Bank|21 Videos

Similar Questions

Explore conceptually related problems

If the chord of contact of tangents from a point P to the parabola y^2=4a x touches the parabola x^2=4b y , then find the locus of Pdot

Find the points of contact Q and R of a tangent from the point P(2,3) on the parabola y^2=4xdot

Prove that the chords of contact of pairs of perpendicular tangents to the ellipse x^2/a^2+y^2/b^2=1 touch another fixed ellipse.

The chords of contact of the pair of tangents drawn from each point on the line 2x + y=4 to the circle x^2 + y^2=1 pass through the point (a,b) then 4(a+b) is

The locus of the middle points of the focal chords of the parabola, y^2=4x is:

The angle between a pair of tangents drawn from a point P to the hyperbola y^2 = 4ax is 45^@ . Show that the locus of the point P is hyperbola.

If the chord of contact of the tangents drawn from a point on the circle x^2+y^2=a^2 to the circle x^2+y^2=b^2 touches the circle x^2+y^2=c^2 , then prove that a ,b and c are in GP.

Find the equation of tangents drawn to the parabola y=x^2-3x+2 from the point (1,-1)dot

The locus of mid-points of a focal chord of the ellipse x^2/a^2+y^2/b^2=1

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