Answer

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Hint- Here, we will be using distance formula.

Given, we have two points A$\left( {2,3} \right)$ and B$\left( { - 4,1} \right)$.

Since, we know that any point on the y-axis will have its x-coordinate as zero.

Let P$\left( {0,y} \right)$be that point that will lie on the y-axis.

We know that according to distance formula, the distance between any two points ${\text{A}}\left( {a,b} \right)$ and $B\left( {c,d} \right)$is given by $d = \sqrt {{{\left( {c - a} \right)}^2} + {{\left( {d - b} \right)}^2}} $

Also given that the point P$\left( {0,y} \right)$is equidistant from the points A$\left( {2,3} \right)$ and B$\left( { - 4,1} \right)$ which means that the distances AP and BP are equal.

i.e. $

{\text{AP}} = {\text{BP}} \Rightarrow \sqrt {{{\left( {0 - 2} \right)}^2} + {{\left( {y - 3} \right)}^2}} = \sqrt {{{\left[ {0 - \left( { - 4} \right)} \right]}^2} + {{\left( {y - 1} \right)}^2}} \\

\Rightarrow \sqrt {4 + {y^2} + 9 - 6y} = \sqrt {16 + {y^2} + 1 - 2y} \\

$

Now squaring both the sides of above equation, we get

$

\Rightarrow 4 + {y^2} + 9 - 6y = 16 + {y^2} + 1 - 2y \Rightarrow 6y - 2y = 4 + 9 - 1 - 16 \\

\Rightarrow 4y = - 4 \Rightarrow y = - 1 \\

$

Therefore, the required point on the y-axis is P$\left( {0, - 1} \right)$.

Note- In this problem if the point which is equidistant from the two given points instead of lying on y-axis, lies on x-axis then the coordinates of the required point would have been assumed as P$\left( {x,0} \right)$ because any point lying on the x-axis have its y coordinate as zero.

Given, we have two points A$\left( {2,3} \right)$ and B$\left( { - 4,1} \right)$.

Since, we know that any point on the y-axis will have its x-coordinate as zero.

Let P$\left( {0,y} \right)$be that point that will lie on the y-axis.

We know that according to distance formula, the distance between any two points ${\text{A}}\left( {a,b} \right)$ and $B\left( {c,d} \right)$is given by $d = \sqrt {{{\left( {c - a} \right)}^2} + {{\left( {d - b} \right)}^2}} $

Also given that the point P$\left( {0,y} \right)$is equidistant from the points A$\left( {2,3} \right)$ and B$\left( { - 4,1} \right)$ which means that the distances AP and BP are equal.

i.e. $

{\text{AP}} = {\text{BP}} \Rightarrow \sqrt {{{\left( {0 - 2} \right)}^2} + {{\left( {y - 3} \right)}^2}} = \sqrt {{{\left[ {0 - \left( { - 4} \right)} \right]}^2} + {{\left( {y - 1} \right)}^2}} \\

\Rightarrow \sqrt {4 + {y^2} + 9 - 6y} = \sqrt {16 + {y^2} + 1 - 2y} \\

$

Now squaring both the sides of above equation, we get

$

\Rightarrow 4 + {y^2} + 9 - 6y = 16 + {y^2} + 1 - 2y \Rightarrow 6y - 2y = 4 + 9 - 1 - 16 \\

\Rightarrow 4y = - 4 \Rightarrow y = - 1 \\

$

Therefore, the required point on the y-axis is P$\left( {0, - 1} \right)$.

Note- In this problem if the point which is equidistant from the two given points instead of lying on y-axis, lies on x-axis then the coordinates of the required point would have been assumed as P$\left( {x,0} \right)$ because any point lying on the x-axis have its y coordinate as zero.

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