Home
Class 12
MATHS
If the chords of contact of tangents fro...

If the chords of contact of tangents from two poinst `(x_1, y_1)` and `(x_2, y_2)` to the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` are at right angles, then find the value of `(x_1x_2)/(y_1y_2)dot`

Text Solution

Verified by Experts

The correct Answer is:
`-a^(4)//b^(4)`
The equations of the chords of contact of tangents drawn from `(x_(1),y_(1)) and (x_(2),y_(2))` to the ellipse
`(x^(2))/(b^(2))+(y^(2))/(b^(2))=1`
are `(x x_(1))/(a^(2))+(yy_(1))/(b^(2))=1" "(1)`
`(x x_(2))/(a^(2))+(yy_(2))/(b^(2))=1" "(2)`
It is given that (1) and (2) are at right angles. Therefore, `(-b^(2))/(a^(2))(x_(1))/(y_(1))xx(-b^(2))/(a^(2))(y_(2))/(y_(2))=-1`
or `(x_(1)y_(1))/(y_(1)y_(2))=-(a^(2))/(b^(4))`
Promotional Banner

Similar Questions

Explore conceptually related problems

If the chords of contact of tangents from two points (-4,2) and (2,1) to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 are at right angle, then find then find the eccentricity of the hyperbola.

Find the maximum area of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 which touches the line y=3x+2.

If the chord of contact of tangents from a point (x_1,y_1) to the circle x^2+y^2=a^2 touches the circle (x-a)^2+y^2=a^2 , then the locus of (x_1,y_1) is :

Find the lengths of the tangents drawn from the point. (x_(1),y_(1)) to the circle x^(2)+y^(2)+2x=0

Find the locus of middle points of chords of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 which subtend right angle at its center.

Two tangents are drawn from a point on hyperbola x^(2)-y^(2)=5 to the ellipse (x^(2))/(9)+(y^(2))/(4)=1 . If they make angle alpha and beta with x-axis, then

Tangents are drawn from any point on the circle x^(2)+y^(2) = 41 to the ellipse (x^(2))/(25)+(y^(2))/(16) =1 then the angle between the two tangents is

If the chord of contact of tangents 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 a, b, c are in-

If the foci of the ellipse (x^2)/(16)+(y^2)/(b^2)=1 and the hyperbola (x^2)/(144)-(y^2)/(81)=1/(25) coincide, then find the value of b^2

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.