Question

Find the condition that :

the line 4x + 5y = 0 coincides with one of the lines given by ax² + 2hxy + by² = 0.

Answer 1

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

Given that 4x + 5y = 0 is one of the lines represented by ax² + 2hxy + by² = 0.

The slope of the line 4x + 5y = 0 is -4/5

∴ m = -4/5 is a root of the auxiliary equation 

bm² + 2hm + a = 0.

∴ 16b – 40h + 25a = 0

∴ 25a + 16b = 40k.

This is the required condition.