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Question
Show that the combined equation of the pair of lines passing through the origin and each making an angle α with the line x + y = 0 is x2 + 2(sec 2α)xy + y2 = 0
Solution
Let OA and OB be the required lines.
Let OA (or OB) has slope m.
∴ its equation is y = mx ...(1)
It makes an angle α with x + y = 0 whose slope is - 1.
∴ tan α = `|("m" + 1)/(1 + "m"(- 1))|`
Squaring both sides, we get,
`"tan"^2alpha = ("m" + 1)^2/(1 - "m")^2`
∴ tan2α (1 - 2m + m2) = m2 + 2m + 1
∴ tan2α - 2mtan2α + m2tan2α = m2 + 2m + 1
∴ (tan2α - 1)m2 - 2(1 + tan2α)m + (tan2α - 1) = 0
∴ `"m"^2 - 2((1 + "tan"^2alpha)/("tan"^2alpha - 1))"m" + 1 = 0`
∴ `"m"^2 + 2((1 + "tan"^2alpha)/(1 - "tan"^2alpha)) "m" + 1 = 0`
∴ `"m"^2 + 2(sec 2 alpha)"m" + 1 = 0` ...`[because "cos 2"alpha = (1 - "tan"^2 alpha)/(1 + "tan"^2alpha)]`
∴ `"y"^2/"x"^2 + 2("sec"2alpha)"y"/"x" + 1 = 0`
∴ `"y"^2 2"xy" "sec" 2 alpha + "x"^2 = 0` ...[By (1)]
∴ `"y"^2 + 2"xy" "sec 2" alpha + "x"^2 = 0`
∴ x2 + 2(sec 2α)xy + y2 = 0 is the required equation.
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