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Question
Prove that one of the straight lines given by ax2 + 2hxy + by2 = 0 will bisect the angle between the coordinate axes if (a + b)2 = 4h2
Solution
The equation of the given pair of straight lines is
ax2 + 2hxy + by2 = 0 ………(1)
Let m1 and m2, be the slopes of the separate straight lines.
Given that one of the straight lines of (1) bisects the angle between the coordinate axes.
∴ The angle made by that line with x-axis 45°.
Slope of that line m1 = tan 45°
m1 = 1
m1 = m2 = `- (2"h")/"b"`, m1m2 = `"a"/"b"`
1 + m2 = `- (2"h")/"b"` (1) m2 = `"a"/"b"`
1 + m2 = `- (2"h")/"b"` m2 = `"a"/"b"`
`1 + "a"/"b" = - (2"h")/"b"`
`("b" + "a")/"b" - (2"h")/"b"`
a + b = – 2h
Squaring on both sides
(a + b)2 = (– 2h)2
(a + b)2 = 4h2
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