Question

Find the condition that any straight line lx + my + n = 0, may touch the conic ax² + 2hxy + by² + 2gx + 2fy + c = 0 

Answer 1

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

ax² + 2hxy + by² + 2gx + 2fy + c = 0 ...(2)

Substituting for y in (2) from (1), we have for the equation giving the abscissa of the points of intersection of (1) and (2),

If (1) be a tangent, the values of x given by (3) must be equal.

The condition for this is,

On simplifying, we have, after division by m²,