Question

Find a point on y-axis which is equidistant from the points $(5,-2)$ and $(-3,2)$.

Answer 1

Find a point on y-axis which is equidistant from the points $(5,-2)$ and $(-3,2)$:

Let the point be $P(0,a)$ be equidistant from the points $A(5,-2)$ and $B(-3,2)$.

$\Rightarrow PA=PB$.

According to the distance formula, the distance between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given by:

$AB=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Let's use the distance formula to substitute the given coordinates in $PA=PB$:

$\Rightarrow$ $\sqrt{{(5-0)}^2+{(-2-a)}^2}$ $=\sqrt{{(-3-0)}^2+{(2-a)}^2}$

$\Rightarrow$ ${(5-0)}^2+{(-2-a)}^2$ $={(-3-0)}^2+{(2-a)}^2$ [ squaring both the sides ]

$\Rightarrow$ $25+{4+4a+a}^2$ $=9+{4-4a+a}^2$

$\Rightarrow$ $29+4a$ $=13-4a$

$\Rightarrow$ $29-13$ $=-4a-4a$

$\Rightarrow$ $16$ $=-8a$

$\Rightarrow$ $a$ $=-2$

Hence, $(0,-2)$ is a point on y-axis which is equidistant from the points $(5,-2)$ and $(-3,2)$.