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Question
If one of the lines given by ax2 + 2hxy + by2 = 0 bisect an angle between the coordinate axes, then show that (a + b)2 = 4h2 .
Solution
The auxiliary equation of the lines given by ax2 + 2hxy + by2 = 0 is bm2 + 2hm + a = 0.
Since one of the lines bisects an angle between the coordinate axes, that line makes an angle of 45° or 135° with the positive direction of X-axis.
∴ the 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.
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