Relation Between Equidistant Points

The problem asks for the relation between the coordinates x and y for a point P(x, y) that is the same distance from two given points, A(5, 3) and B(4, 6).

We use the distance formula, which states that the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

The condition is that the distance PA must equal the distance PB.

PA = $\sqrt{(x - 5)^2 + (y - 3)^2}$

PB = $\sqrt{(x - 4)^2 + (y - 6)^2}$

Equating Distances

Since PA = PB, we can equate their squares to simplify the calculation and remove the square roots:

PA$^2$ = PB$^2$

$(x - 5)^2 + (y - 3)^2 = (x - 4)^2 + (y - 6)^2$

Expanding and Simplifying

Expand the squared terms:

$(x^2 - 10x + 25) + (y^2 - 6y + 9) = (x^2 - 8x + 16) + (y^2 - 12y + 36)$

Cancel out the $x^2$ and $y^2$ terms from both sides:

$-10x + 25 - 6y + 9 = -8x + 16 - 12y + 36$

Combine constant terms on each side:

$-10x - 6y + 34 = -8x - 12y + 52$

Finding the Relation

Rearrange the equation to group terms with x, y, and constants:

Bring x terms to the right side: $-8x + 10x = 2x$

Bring y terms to the left side: $-6y + 12y = 6y$

Bring constant terms to the right side: $52 - 34 = 18$

The equation becomes:

$6y - 2x = 18$

Divide the entire equation by 2 to simplify:

$3y - x = 9$

This is the required relation between x and y.