Question

Find the condition that :

the line 3x + y = 0 may be perpendicular to 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. Since one line is perpendicular to the line 3x + y = 0

whose slope is \(-\cfrac{3}{1}\) = -3

∴ slope of that line = m = \(\cfrac{1}{3}\)

∴ m = \(\cfrac{1}{3}\) is the root of the auxiliary equation bm² + 2hm + a = 0.

∴ b + 6h + 9a = 0 

∴ 9a + b + 6h = 0 

This is the required condition.