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Question: The lines joining the points of intersection of line \(x + y = 1\) and curve \(x^{2} + y^{2} - 2y + ...

The lines joining the points of intersection of line x+y=1x + y = 1 and curve x2+y2−2y+λ=0x^{2} + y^{2} - 2y + \lambda = 0 to the origin are perpendicular, then the value of 1/101/\sqrt{10} will be

A

1/2

B

–1/2

C

1/21/\sqrt{2}

D

0

Answer

0

Explanation

Solution

Making the equation of curve homogeneous with the help of line x+y=1x + y = 1, we get x2+y2−2y(x+y)+λ(x+y)2=0x^{2} + y^{2} - 2y(x + y) + \lambda(x + y)^{2} = 0

⇒x2(1+λ)+y2(−1+λ)−2yx=0\Rightarrow x^{2}(1 + \lambda) + y^{2}( - 1 + \lambda) - 2yx = 0

Therefore the lines be perpendicular, if A+B=0A + B = 0.

⇒1+λ−1+λ=0⇒λ=0\Rightarrow 1 + \lambda - 1 + \lambda = 0 \Rightarrow \lambda = 0.