Question

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). ${x^2} + {y^2} - 4|x| = 0$

Answer 1

Locus is a Latin word for location. Locus can be defined as the set of all the points, which can be anything a line or a curve or a surface, whose location is determined by one or more conditions. Distance from x-axis is measured on y-axis and distance from y-axis is measured on x-axis. Formulae of distance from origin and x-axis, y-axis are to be used here in finding out locus.

Complete step-by-step solution:
Let the point be named A (m, n)
It is asked that 4 times the distance of A from X-axis is equal to the square of its distance from origin.
Equation interpreted from the question:
$4 \times$ distance of A from X-axis = $\text{distance from origin}^2$ -----$\left(1\right)$
We know that distance of any point from the origin, i.e. the distance formula is given by $\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}}$
Therefore, distance of A from origin (0, 0) = $\sqrt {{{(m - 0)}^2} + {{(n - 0)}^2}} = \sqrt {{m^2} + {n^2}}$-------$\left(2\right)$
Distance of any point from the x-axis = $\sqrt {{{(y - 0)}^2}} = |y|$
Therefore, distance of A from x-axis = $\sqrt {{{(n - 0)}^2}} = |n|$
Therefore, 4 x distance of A from X-axis = $4 \times |n|$---3
Therefore, equation 1 becomes,
$4 \times |n| = {(\sqrt {{m^2} + {n^2}} )^2} = {m^2} + {n^2}$, using equation 2 and 3
To obtain the locus of the point A (m, n), we have to just replace m and n from the above equation with x and y respectively.
The equation after replacing the values becomes and taking $4 \times |n|$ on other side, we get
${x^2} + {y^2} - 4|y| = 0$
Therefore, ${x^2} + {y^2} - 4|y| = 0$ is the locus to be which was required.
Hence, from the given multiple choices – the B option is the correct answer.

Note: Concept of locus, i.e. what is locus and its formula has to be known. Also the formulae of distance are to be known. Also, modulus is the important part of the answer, as you can see the similar option but without the modulus of y. After taking the square root, modulus has to be applied, since after taking the mod sign, even the negative values become positive. Equation for locus has to be in terms x and y.