Question

Find the equations to the tangents to the conic x² + 4xy + 3y² - 5x - 6y + 3 = 0 which are parallel to the straight line x + 4y = 0.

Answer 1

x² + 4xy + 3y² - 5x - 6y + 3 = 0...(1)

The equation to any such tangent is

x + 4y + c = 0.........(2),

where c is to be determined.

This straight line meets (1) in points given by

The roots of this equation are equal, i.e. the line (2) is a tangent, if

The required tangents are therefore