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From a point P tangents are drawn to the ellipse x^2+a^2+y^2+b^2=1.If the chord of contact touches the auxiliary circle,then the locus of P

From a point P tangents are drawn to the ellipse x^2+a^2+y^2+b^2=1.If the chord of contact touches the auxiliary circle,then the locus of P 

Grade:11

1 Answers

Saurabh Koranglekar
askIITians Faculty 10336 Points
2 years ago

Let the locus of p be (h,k)So eqn. Of chord of contact to the ellipse will be xh/a^2+yk/b^2=1.Since this chord of contact is tangent to the auxillary circle,=> it`s distance from the center of the circle {I.e.(0,0)} will be the radius of the auxillary circle , I.e. `a` .Using this,|-a^2b^2|/√[(h^2)(b^4)+(k^2)(a^4)]=a=> (a^2) (b^4)=(h^2)(b^4)+(k^2)(a^4)Dividing throughout by (a^4)(b^4) and replacing h and k by x and y respectively gives us the required locus,I.e. (x^2/a^4)+(y^2/b^4)=1/(a^2)

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