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

Given:

Given points are $(-5, -2)$ and $(3, 2)$.

To do:

We have to find the point(s) on y-axis which is equidistant from $(-5, -2)$ and $(3, 2)$.

Solution:

Let the coordinates of the two points be $A (-5, -2)$ and $B (3, 2)$.

We know that,

The x coordinate of a point on the y-axis is $0$.
Let the coordinates of the point which is equidistant from the points $A$ and $B$ be $C(0, y)$.

This implies,

$AC = CB$

The distance between two points $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\mathrm{B}\left(x_{2}, y_{2}\right)$ is $\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$.

Therefore,

$AC=\sqrt{(0+5)^2+(y+2)^2}$

$=\sqrt{25+(y+2)^2}$

$CB=\sqrt{(0-3)^{2}+(y-2)^{2}}$

$=\sqrt{9+(y-2)^{2}}$

$\Rightarrow \sqrt{25+(y+2)^{2}}=\sqrt{9+(y-2)^{2}}$

Squaring on both sides, we get,

$25+(y+2)^{2}=9+(y-2)^{2}$

$y^{2}+4 y+4+25=y^{2}-4 y+4+9$

$4 y+4 y=9-25$

$8y=-16$

$\Rightarrow y=\frac{-16}{8}$

$y=-2$

Therefore, the required point is $(0, -2)$.

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