Question

Find the locus of a point whose distance from the point $(1, 2)$ is equal to its distance from the axis of y.

Answer 1

Hint: First suppose the coordinate of the points. Then use the distance formula to obtain the distance between the point and $(1, 2)$ and the y-axis. Then equate the obtained distances to obtain the required result.

Formula Used:
The distance formula of two points $(a,b),(c,d)$ is
$\sqrt {{{(c - a)}^2} + {{(d - b)}^2}} $ .

Complete step by step solution:
Suppose that the coordinate of the point is $P(h,k)$ and suppose $(1, 2)$ is the point A.
It is given that $PA = h$ --(1)
Now,
$PA = \sqrt {{{(1 - h)}^2} + {{(2 - k)}^2}} $
Therefore, from equation (1) we have,
$\sqrt {{{(1 - h)}^2} + {{(2 - k)}^2}} = h$
Square both sides of the equation,
${(1 - h)^2} + {(2 - k)^2} = {k^2}$
$1 - 2h + {h^2} + 4 - 4k + {k^2} = {h^2}$
${k^2} - 2h - 4k + 5 = 0$
Therefore, the locus is ${y^2} - 2x - 4y + 5 = 0$.

Note: Here The concept of locus, i.e. what is locus and its formula, must be understood. Distance formulae must also be known. Also equation for locus has to be in terms x and y. Remember that the distance of a point from the x axis is always the y coordinate and the distance from the y axis is always the x coordinate, no need to calculate these results.