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Locus of the point ' P ', such that sum of squares of it's distances from lines \mathrm{2 x^2+y^2-3 x y=0} is 1 , is
 

Option: 1

 a straight line passing through origin
 


Option: 2

 an ellipse centered at origin
 


Option: 3

 a parabola having vertex at origin
 


Option: 4

 none of these
 


Answers (1)

best_answer

Equation of lines \mathrm{ 2 x^2+y^2-3 x y=0}
\mathrm{ \Rightarrow 2 x-y=0\ or \ x-y=0}

Let \mathrm{ P(h, k)} be any point.

According given condition
\begin{aligned} &\mathrm{ \left(\frac{2 h-k}{\sqrt{5}}\right)^2+\left(\frac{h-k}{\sqrt{2}}\right)^2=1} \\ & \mathrm{\Rightarrow 13 h^2+7 k^2-18 h k-10=0 }\end{aligned}
Locus is \mathrm{ 13 x^2+7 y^2-18 x y-10=0}
\Rightarrow An ellipse center at O(0,0) (since, \mathrm{ \Delta \neq 0, H^2<A B} )

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Anam Khan

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