The locus of a point whose chord of contact with respect to the ellipse x2+2y2=1 subtends a right angle at the centre of the ellipse is
Dear Student,
Please find below the solution to the asked query :
ellipse is x^2+ 2y^2 = 1 ---------------------------------------(...
let (h,k,) is moving point
equation to the chord of contact is ;--
hx + 2ky =1--------------------------(II)
to get theequation of line joining the center (0.0 ) of the ellipse to
the point of intersection of (I) 7 (II) make (I) homogeneous with
the help of (II)
x^2 + 2y^2 = ( hx+ 2ky)^2
or x^2 + 2y^2= h^2x^2+ 4hkxy+ 4k^2y^2
x^2[ 1- h^2) + y^2[ 2- 4k^2) - 4hkxy =0
the lines are at right angle if ---
coefficient of x^2+ coefficient of Y^2 = 0
1- h^2 + 2- 4k^2 = 0
or h^2 + 4k^2 = 3
so locus of (h,k ) is ;--[putting h=x & k=y ]
x^2 + 4y^2 = 3
Hope this would clear your doubt about the topic.
If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible.
Regards
Please find below the solution to the asked query :
ellipse is x^2+ 2y^2 = 1 ---------------------------------------(...
let (h,k,) is moving point
equation to the chord of contact is ;--
hx + 2ky =1--------------------------(II)
to get theequation of line joining the center (0.0 ) of the ellipse to
the point of intersection of (I) 7 (II) make (I) homogeneous with
the help of (II)
x^2 + 2y^2 = ( hx+ 2ky)^2
or x^2 + 2y^2= h^2x^2+ 4hkxy+ 4k^2y^2
x^2[ 1- h^2) + y^2[ 2- 4k^2) - 4hkxy =0
the lines are at right angle if ---
coefficient of x^2+ coefficient of Y^2 = 0
1- h^2 + 2- 4k^2 = 0
or h^2 + 4k^2 = 3
so locus of (h,k ) is ;--[putting h=x & k=y ]
x^2 + 4y^2 = 3
Hope this would clear your doubt about the topic.
If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible.
Regards