The locus of the points of intersection of the tangents at the extremities of the chords of the ellipse x^2+2y^2=6 - Maths - Conic Sections

Question

The locus of the points of intersection of the tangents at
the extremities of the chords of the ellipse x^2+2y^2=6 which touch
the ellipse ​x^2+4y^2=4

Neha Sethi · Accepted Answer

Dear student
We can writex2+4y2=4 asx24+y21=1      
(1)Equation of a tangent to the ellipse (1) isx2cosθ+ysinθ=1     
(2)Equation of the ellipse x2+2y2=6 can be written asx26+y23=1     
(3)Suppose (2) meets the ellipse (3) at P and Q and the tangents at P and Q to the ellipse (3) intersect at (h,k),then (2) is the chord of contact of (h,k) with respect to theellipse (3) and thus its equation ishx6+ky3=1     
(4)Since (2) and (4) represent the same lineh6cosθ/2=k3sinθ=1⇒h=3cosθ, k=3sinθand the locus of (h,k) is x2+y2=9
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The locus of the points of intersection of the tangents at the extremities of the chords of the ellipse x^2+2y^2=6 which touch the ellipse ​x^2+4y^2=4.

The locus of the points of intersection of the tangents at the extremities of the chords of the ellipse x^2+2y^2=6 which touch the ellipse x^2+4y^2=4.