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Question
If the lines represented by ax2 + 2hxy + by2 = 0 make angles of equal measure with the coordinate axes, then show that a ± b.
OR
Show that, one of the lines represented by ax2 + 2hxy + by2 = 0 will make an angle of the same measure with the X-axis as the other makes with the Y-axis, if a = ± b.
Solution
Let OA and OB be the two lines through the origin represented by ax2 + 2hxy + by2 = 0.
Since these lines make angles of equal measure with the coordinate axes, they make angles α and `pi/2 - alpha` with the positive direction of X-axis or α and `pi/2 + alpha` with the positive direction of X-axis.
∴ slope of the line OA = m1 = tan α
and slope of the line OB = m2
= `"tan"(pi/2 - alpha) "or" "tan"(pi/2 + alpha)`
i.e. m2 = cot α or m2 = - cot α
∴ m1m2 = tan α × cot α = 1
OR m1m2 = tan α (- cot α) = - 1
i.e. m1m2 = ± 1
But m1m2 = `"a"/"b"`
∴ `"a"/"b" = +- 1`
∴ a = ± b
This is the required condition.
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