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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 . - Mathematics and Statistics

Sum

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 .

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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.

Concept: Homogeneous Equation of Degree Two
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