# The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from \[Q\left( {2, - 5} \right)\] and \[R\left( { - 3,6} \right)\], then find the coordinates of P.

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Hint: Use the distance formula $D = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $ to compare the distance of P from two points and find the coordinates.

Complete step-by-step answer:

Let the coordinate of point P is $\left( {x,y} \right)$.

According to the question, the x-coordinate of P is twice its y-coordinate. Then we have:

$ \Rightarrow x = 2y$

So, the coordinates of P will be $\left( {2y,y} \right)$.

We know that the distance between two point $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ can be found using distance formula:

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

Using this, the distance between points $P\left( {2y,y} \right)$ and point \[Q\left( {2, - 5} \right)\] is:

$ \Rightarrow PQ = \sqrt {{{\left( {2y - 2} \right)}^2} + {{\left( {y + 5} \right)}^2}} $

And the distance between points $P\left( {2y,y} \right)$ and point \[R\left( { - 3,6} \right)\] is:

$ \Rightarrow PR = \sqrt {{{\left( {2y - 3} \right)}^2} + {{\left( {y + 6} \right)}^2}} $

Given in the question, P is equidistant from Q and R. So, we have:

$

\Rightarrow PQ = PR \\

\Rightarrow \sqrt {{{\left( {2y - 2} \right)}^2} + {{\left( {y + 5} \right)}^2}} = \sqrt {{{\left( {2y - 3} \right)}^2} + {{\left( {y + 6} \right)}^2}} \\

\Rightarrow 4{y^2} + 4 - 8y + {y^2} + 25 + 10y = 4{y^2} + 9 - 12y + {y^2} + 36 + 12y \\

\Rightarrow 2y + 29 = 45 \\

\Rightarrow 2y = 16 \\

\Rightarrow y = 8 \\

$

So, the coordinate of point P is $\left( {16,8} \right)$.

Note: In the above question, if in such cases the slope PQ and PR also comes out to be equal then, P, Q and R will lie on the same straight line i.e. they’ll be collinear and P will be the mid-point of Q and R.

Complete step-by-step answer:

Let the coordinate of point P is $\left( {x,y} \right)$.

According to the question, the x-coordinate of P is twice its y-coordinate. Then we have:

$ \Rightarrow x = 2y$

So, the coordinates of P will be $\left( {2y,y} \right)$.

We know that the distance between two point $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ can be found using distance formula:

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

Using this, the distance between points $P\left( {2y,y} \right)$ and point \[Q\left( {2, - 5} \right)\] is:

$ \Rightarrow PQ = \sqrt {{{\left( {2y - 2} \right)}^2} + {{\left( {y + 5} \right)}^2}} $

And the distance between points $P\left( {2y,y} \right)$ and point \[R\left( { - 3,6} \right)\] is:

$ \Rightarrow PR = \sqrt {{{\left( {2y - 3} \right)}^2} + {{\left( {y + 6} \right)}^2}} $

Given in the question, P is equidistant from Q and R. So, we have:

$

\Rightarrow PQ = PR \\

\Rightarrow \sqrt {{{\left( {2y - 2} \right)}^2} + {{\left( {y + 5} \right)}^2}} = \sqrt {{{\left( {2y - 3} \right)}^2} + {{\left( {y + 6} \right)}^2}} \\

\Rightarrow 4{y^2} + 4 - 8y + {y^2} + 25 + 10y = 4{y^2} + 9 - 12y + {y^2} + 36 + 12y \\

\Rightarrow 2y + 29 = 45 \\

\Rightarrow 2y = 16 \\

\Rightarrow y = 8 \\

$

So, the coordinate of point P is $\left( {16,8} \right)$.

Note: In the above question, if in such cases the slope PQ and PR also comes out to be equal then, P, Q and R will lie on the same straight line i.e. they’ll be collinear and P will be the mid-point of Q and R.

Last updated date: 25th Sep 2023

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