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Question
Prove that the product of length of perpendiculars drawn from P(x1, y1) to the lines represented by ax2 + 2hxy + by2 = 0 is `|("ax"_1^2 + "2hx"_1"y"_1 + "by"_1^2)/(sqrt("a - b")^2 + "4h"^2)|`
Solution
Let m1 and m2 be the slopes of the lines represented by ax2 + 2hxy + by2 = 0
∴ m1 + m2 = `- "2h"/"b"` and m1m2 = `"a"/"b"` ...(1)
The separate equations of the lines represented by ax2 + 2hxy + by2 = 0 are
y = m1x and y = m2x
i.e. m1x - y = 0 and m2x - y = 0
Length of perpendicular from P(x1, y1) on
m1x - y = 0 is `|("m"_1"x"_1 - "y"_1)/(sqrt("m"_1^2 + 1))|`
Length of perpendicular form P(x1, y1) on
m2x - y = 0 is `|("m"_2"x"_1 - "y"_1)/(sqrt("m"_2^2 + 1))|`
∴ product of lengths of perpendiculars
`= |("m"_1"x"_1 - "y"_1)/(sqrt("m"_1^2 + 1))| xx |("m"_2"x"_1 - "y"_1)/(sqrt("m"_2^2 + 1))|`
`= |("m"_1"m"_2"x"_1^2 - ("m"_1 + "m"_2)"x"_1"y"_1 + "y"_1^2)/(sqrt("m"_1^2"m"_2^2 + "m"_1^2 + "m"_2^2 + 1))|`
`= ("m"_1"m"_2"x"_1^2 - ("m"_1 + "m"_2)"x"_1"y"_1 + "y"_1^2)/(sqrt("m"_1^2"m"_2^2 + ("m"_1 + "m"_2)^2 - "2m"_1"m"_2 + 1)`
`= |("a"/"b"."x"_1^2 - (- "2h")/"b" "x"_1"y"_1 + "y"_1^2)/(sqrt("a"^2/"b"^2 + (- "2h")/"b" - "2a"/"b" + 1))|` ...(By (1))
`= |("ax"_1^2 + "2hx"_1"y"_1 + "by"_1^2)/(sqrt("a"^2 + 4"h"^2 - "2ab" + "b"^2))|`
`= |("ax"_1^2 + "2hx"_1"y"_1 + "by"_1^2)/(sqrt(("a"^2 - "2ab" + "b"^2) + "4h"^2))|`
`= |("ax"_1^2 + "2hx"_1"y"_1 + "by"_1^2)/(sqrt(("a - b")^2 + "4h"^2))|`
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