Question

Find the relation between \( x \) and \( y \) such that the point \( (x, y) \) is equidistant from the points (3, 6) and (-3, 4).

Hint:

For points equidistant from two fixed points, use the distance formula and square both sides to eliminate the square root.

Answer 1

Step 1: Use the concept of equal distances.
\[ \sqrt{(x - 3)^2 + (y - 6)^2} = \sqrt{(x + 3)^2 + (y - 4)^2} \]
Step 2: Square both sides.
\[ (x - 3)^2 + (y - 6)^2 = (x + 3)^2 + (y - 4)^2 \]
Step 3: Expand and simplify.
\[ x^2 - 6x + 9 + y^2 - 12y + 36 = x^2 + 6x + 9 + y^2 - 8y + 16 \] Cancel \( x^2, y^2, 9 \): \[ -6x - 12y + 36 = 6x - 8y + 16 \] \[ -12x - 4y + 20 = 0 \] \[ 3x + y - 5 = 0 \]
Step 4: Final Relation.
\[ \boxed{3x + y - 5 = 0} \]