Locus Problem #2

Geometry Level 4

An ellipse with semi-major axis a a and semi-minor axis b b touches both the coordinate axis , then locus of its centre is a part of the curve :

x y = a 2 + b 2 xy=a^2+b^2 x 2 + y 2 = a 2 + b 2 x^2+y^2=a^2+b^2 1 x 2 + 1 y 2 = a 2 + b 2 \dfrac{1}{x^2}+ \dfrac{1}{y^2}=a^2+b^2 x 2 y 2 = a 2 + b 2 x^2y^2=a^2+b^2

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1 solution

Use the fact : p 1 2 + p 2 2 = a 2 + b 2 p_{1}^2+p_{2}^2=a^2+b^2 , where p 1 p_{1} and p 2 p_{2} are lengths of perpendiculars from centre of the ellipse on two perpendicular tangents.

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