The locus of a point which moves such that it | vedclass

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(B) Let the point be $P(x, y)$.Given that the distance of $P$ from the origin $O(0, 0)$ is thrice its distance from the line $2x - y = 0$.So,$OP = 3 \

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a pair of straight lines

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(B) Let the point be $P(x, y)$.Given that the distance of $P$ from the origin $O(0, 0)$ is thrice its distance from the line $2x - y = 0$.So,$OP = 3 	imes PM$,where $PM$ is the perpendicular distance from $P$ to the line $2x - y = 0$.$\sqrt{x^2 + y^2} = 3 \frac{|2x - y|}{\sqrt{2^2 + (-1)^2}}$$\sqrt{x^2 + y^2} = 3 \frac{|2x - y|}{\sqrt{5}}$Squaring both sides:$x^2 + y^2 = \frac{9(2x - y)^2}{5}$$5(x^2 + y^2) = 9(4x^2 + y^2 - 4xy)$$5x^2 + 5y^2 = 36x^2 + 9y^2 - 36xy$$31x^2 + 4y^2 - 36xy = 0$This is a homogeneous equation of the second degree in $x$ and $y$,which represents a pair of straight lines passing through the origin.Alternatively,since the focus $(0, 0)$ lies on the directrix $y = 2x$,the conic section degenerates into a pair of straight lines.

The locus of a point which moves such that its distance from the origin is thrice its distance from the line $y = 2x$ is

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