Question

Find the equation to conics which are confocal with the ellipse

x²/a² + y²/b² = 1 

Answer 1

x²/a² + y²/b² = 1 ......(1)

All conics having the same foci have the same centre and axes.

The equation to any conic having the same centre and axes as the given conic is

The foci of (1) are at the points

The foci of (2) are at the points

These foci are the same if

The equation (2) then becomes

which is therefore the required equation, the quantity λ, determining the particular confocal.