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Prove that the locus of feet of perpendiculars from foci upon any tangent to an ellipse is an auxiliary circle

Prove that the locus of feet of perpendiculars from foci upon any tangent to an ellipse is an auxiliary circle

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Grade:12th pass

1 Answers

Arun
25750 Points
4 years ago
We know that equation of any tangent to ellipse is y=mx+(a^2m^2+b^2)^1/2 ..........(1)Any point F (h,k) lies on tangent k-mh=(a^2m^2+b^2)^1/2 .............(2)Focus S (ae,0) slope of line SF = (k-0)/h-aeSF is perpendicular to line(1) m1m2=-1 k-0/h-ae*m =-1 》m=ae-h/k 》km+h=ae... (4)Square individually 3 and 4 equation and addK^2+m^2+k^2m^2+h^2=a^2m^2+b^2+a^2e^2 》k^2 (1+m^2)+h^2 (1+m^2)=a^2 (1+m^2)》h^2+k^2=a^2》therefore locus is x^2+y^2=a^2Which is equation of auxiliary circle

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