(B) Let the coordinates of the moving point be $(x, y)$.The distance of the point $(x, y)$ from the $x$-axis is $|y|$.The distance of the point $(x, y

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(B) Let the coordinates of the moving point be $(x, y)$.The distance of the point $(x, y)$ from the $x$-axis is $|y|$.The distance of the point $(x, y)$ from the origin $(0, 0)$ is $\sqrt{x^2 + y^2}$.According to the problem,$4 \times (\text{distance from } x\text{-axis}) = (\text{distance from origin})^2$.Substituting the values,we get $4|y| = (\sqrt{x^2 + y^2})^2$.This simplifies to $4|y| = x^2 + y^2$.Rearranging the terms,we get $x^2 + y^2 - 4|y| = 0$.

Class 11 Mathematics 10-1.Circle and System of Circles Locus Related Problem

Easy

What is the equation of the locus of a point which moves such that $4$ times its distance from the $x$-axis is the square of its distance from the origin?

A
$x^2 + y^2 - 4y = 0$
B
$x^2 + y^2 - 4|y| = 0$
C
$x^2 + y^2 - 4x = 0$
D
$x^2 + y^2 - 4|x| = 0$

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10-1.Circle and System of Circles

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Locus Related Problem

The locus of the midpoints of the chords of the circle $x^{2}+y^{2}=4$ which subtend a right angle at the origin is

DifficultKCET 2009

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If the tangents drawn from a point $P$ to the circle $x^2 + y^2 = a^2$ are perpendicular to the tangents drawn to the circle $x^2 + y^2 = b^2$,then the locus of $P$ is:

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If ${\theta _1}$ and ${\theta _2}$ are the inclinations of the tangents drawn from a point $P(h, k)$ to the circle ${x^2} + {y^2} = {a^2}$ with the $x$-axis,then the locus of $P$,given that $\cot {\theta _1} + \cot {\theta _2} = c$,is:

Difficult

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The equation to the locus of a point which moves so that its distance from the $x$-axis is always one-half its distance from the origin is:

Medium

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If the locus of the point,whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5:4$,is $ax^2+by^2+cxy+dx+ey+170=0$,then the value of $a^2+2b+3c+4d+e$ is equal to:

DifficultJEE MAIN 2024

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What is the equation of the locus of a point which moves such that $4$ times its distance from the $x$-axis is the square of its distance from the origin?

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The locus of the midpoints of the chords of the circle $x^{2}+y^{2}=4$ which subtend a right angle at the origin is

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