If a point \[P\left( {x,y} \right)\] is equidistant from the points \[A\left( {6, - 1} \right)\] and \[B\left( {2,3} \right)\] , find the relation between x and y.
VerifiedHint: By using the distance formula for the points P and A and the points P and B, find the distances PA and PB respectively. Point P is equidistant from A and B, so compare the distances PA and PB. On solving further, you will get a linear equation which is the relation between x and y.
Complete step-by-step answer:
It is given that the points \[P\left( {x,y} \right)\] is equidistant from the points \[A\left( {6, - 1} \right)\] and \[B\left( {2,3} \right)\].
So, the distance between points P and A is equal to the distance between the points P and B i.e. PA=PB.
Now, we can write the distance formula for the distance between points P and A as \[PA = \sqrt {{{\left( {x - 6} \right)}^2} + {{\left( {y - \left( { - 1} \right)} \right)}^2}} \] and for the distance between points P and B as \[PB = \sqrt {{{\left( {x - 2} \right)}^2} + {{\left( {y - 3} \right)}^2}} \].
We know that,
\[PA = PB\]
\[\therefore \sqrt {{{\left( {x - 6} \right)}^2} + {{\left( {y - \left( { - 1} \right)} \right)}^2}} = \sqrt {{{\left( {x - 2} \right)}^2} + {{\left( {y - 3} \right)}^2}} \]
\[\therefore \sqrt {{{\left( {x - 6} \right)}^2} + {{\left( {y + 1} \right)}^2}} = \sqrt {{{\left( {x - 2} \right)}^2} + {{\left( {y - 3} \right)}^2}} \]
Now, on squaring both sides, we get
\[{\left( {x - 6} \right)^2} + {\left( {y + 1} \right)^2} = {\left( {x - 2} \right)^2} + {\left( {y - 3} \right)^2}\]
On expanding every bracket, we get
\[{x^2} - 12x + 36 + {y^2} + 2y + 1 = {x^2} - 4x + 4 + {y^2} - 6y + 9\]
\[\therefore - 12x + 2y + 4x + 6y = 13 - 37\]
\[\therefore - 8x + 8y = -24\]
\[\therefore 8\left( { - x + y} \right) = -24\]
\[\therefore - x + y = -3\]
\[\therefore x-y = 3\]
Hence, the answer is \[x -y = 3\].
Note: If any point (let’s say P) is equidistant from two other points, then always find the distance of point P from the remaining two individual points and compare both the distances to get the solution.
