Question

Find the envelope of the straight line lx + my + n = 0. where the quantities l, m, and n are connected by the relation al² + bm² + cn² + 2fmn + 2gnl + 2hlm = 0 

Answer 1

lx + my + n = 0......(1)

al² + bm² + cn² + 2fmn + 2gnl + 2hlm = 0 .........(2)

[Equation (1) contains two independent parameters l/n and m/n, whilst (2) is an equation connecting them. We could therefore solve (2) to give l/n in terms of m/n; on substituting in (1) we should then have an equation containing one independent parameter and its envelope could then be found.

It is easier, however, to proceed as follows.]

Eliminating n between (1) and (2), we see that the equation to the straight line may be written in the form

The envelope of this is,

The envelope is therefore a conic section.