Question

The coordinates of the point on y-axis which is equidistant from the points $(3,1)$ and $(1,5)$ are

• A. $(0,4)$
• B. $(0,2)$
• C. $(4,0)$
• D. $(2,0)$

Answer 1

The correct option is B $(0,2)$

Distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ can be calculated using the formula $\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Let the point on the y-axis be $(0,y)$

Distance between $(0,y)$ and $(3,1)=\sqrt{{(3-0)}^2+{(1-y)}^2}=\sqrt{9+1^2+y^2-2y}=\sqrt{y^2-2y+10}$

Distance between $(0,y)$ and $(1,5)=\sqrt{{(1-0)}^2+{(5-y)}^2}=\sqrt{1+5^2+y^2-10y}=\sqrt{y^2-10y+26}$

As the point $(0,y)$ is equidistant from the two points, both the distances calculated are equal.

$\sqrt{y^2-2y+10}=\sqrt{y^2-10y+26}$

$y^2-2y+10=y^2-10y+26$

$10y-2y=26-10$

$8y=16$

$y=2$

Thus, the point is $(0,2)$