Question

Through any point in the plane of a given conic there can he drawn two conics confocal with it; also one of these is an ellipse and the other a hyperbola.

Answer 1

Let the equation to the given conic be

and let the given point be (f, g).

Any conic confocal with the given conic is

If this go through the point (f, g), we have

This is a quadratic equation to determine λ and therefore gives two values of λ.

On applying the criterion, we at once see that the roots of this equation are both real.

Also, since its last term is negative, the product of these roots is negative, and therefore one value of μ, is positive and the other is negative.

The two values of b² + λ are therefore one positive and

the other negative. Similarly, the two values of a² + λ can be shewn to be both positive.

On substituting in (2) we thus obtain an ellipse and a hyperbola.