Question

If the distance of the point P(x,y) from the points A(5,1) and B(-1,5) are equal ,then prove that 3x=2y.

Answer 1

Hint: If coordinate of the point P is\[({x_1},{y_1})\] & Q is\[({x_2},{y_2})\], then the distance between this two point is given by the formula,
\[\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} \]

Complete step by step answer:
It is given that the distance of the point P(x,y) from the points A(5,1) and B(-1,5) are equal.
That is the distance between the points A, P & B, P is the same.
Now, let us find the distance between the point A and point P using the given formula, then we get
\[\sqrt {{{(x - 5)}^2} + {{(y - 1)}^2}} \]
Now let us find the distance between the point B and point P using the formula, then we get
\[\sqrt {{{(x + 1)}^2} + {{(y - 5)}^2}} \]
From the given question we have the distances are equal so let us equate the two distance,
So that we get,
\[\sqrt {{{(x - 5)}^2} + {{(y - 1)}^2}} = \sqrt {{{(x + 1)}^2} + {{(y - 5)}^2}} \]
Now let us square both sides of the above equation, then we get,
\[{(x - 5)^2} + {(y - 1)^2} = {(x + 1)^2} + {(y - 5)^2}\]
Now let us expand the above equation using algebraic identities, then we get,
\[{x^2} - 10x + 25 + {y^2} - 2y + 1 = {x^2} + 2x + 1 + {y^2} - 10y + 25\]
Here we are going to cancel the common term from both sides of the above equation, then we get,
\[ - 10x - 2y = 2x - 10y\]
Now let us group the same variables in the equation, then we get,
\[12x = 8y\]
On solving the above found result, we get
\[3x = 2y\]
Hence we have found the result that 3x=2y.

Therefore we have proved that if the distance of the point P(x,y) from the points A(5,1) and B(-1,5) are equal ,then it is true that 3x=2y.

Note:
We have used the following algebraic identities in the above problem,
$
  {(a + b)^2} = {a^2} + 2ab + {b^2} \\
  {(a - b)^2} = {a^2} - 2ab + {b^2} \\
$
While calculating the distance, you need to take care particularly if there is any negative sign between the coordinates of a point.