Question

If one of the lines given by ax² + 2hxy + by² = 0 bisects an angle between co-ordinate axes then show that (a + b)² = 4h².

Answer 1

The auxiliary equation of the lines given by ax² + 2hxy + by² = 0 is bm² + 2hm + a = 0.

Since one of the line bisects an angle between the coordinate axes, that line makes an angle of 45° or 135° with the positive direction of X-axis.

∴ slope of that line = tan 45° or tan 135° 

∴ m = tan 45° = 1

or m = tan 135° = tan (180° – 45°)

= -tan 45°= -1

∴ m = ±1 are the roots of the auxiliary

equation bm² + 2hm + a = 0.

∴ b(±1)² + 2h(±1) + a = 0

∴ b ± 2h + a = 0

∴ a + b = ± 2h

∴ (a + b)² = 4h²

This is the required condition.