Question

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

Hint:

To find the relation between \( x \) and \( y \) for equidistant points, set the distances equal to each other, square both sides, and solve the resulting equation.

Answer 1

The condition that the point \( (x, y) \) is equidistant from the points \( (3, 6) \) and \( (-3, 4) \) means that the distance between \( (x, y) \) and \( (3, 6) \) is equal to the distance between \( (x, y) \) and \( (-3, 4) \). Using the distance formula, the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. \]
Step 1: The distance between \( (x, y) \) and \( (3, 6) \) is: \[ \sqrt{(x - 3)^2 + (y - 6)^2}. \] The distance between \( (x, y) \) and \( (-3, 4) \) is: \[ \sqrt{(x + 3)^2 + (y - 4)^2}. \] Since the distances are equal, we have: \[ \sqrt{(x - 3)^2 + (y - 6)^2} = \sqrt{(x + 3)^2 + (y - 4)^2}. \]
Step 2: Square both sides to eliminate the square roots: \[ (x - 3)^2 + (y - 6)^2 = (x + 3)^2 + (y - 4)^2. \] Expand both sides: \[ (x^2 - 6x + 9) + (y^2 - 12y + 36) = (x^2 + 6x + 9) + (y^2 - 8y + 16). \] Simplify: \[ x^2 - 6x + 9 + y^2 - 12y + 36 = x^2 + 6x + 9 + y^2 - 8y + 16. \] Cancel the \( x^2 \) and \( y^2 \) terms from both sides: \[ -6x + 9 + 36 - 12y = 6x + 9 - 8y + 16. \] Simplify further: \[ -6x + 45 - 12y = 6x + 25 - 8y. \] Move all terms involving \( x \) and \( y \) to one side: \[ -6x - 6x + 45 - 12y + 8y = 25. \] Simplify: \[ -12x - 4y + 45 = 25 \quad \Rightarrow \quad -12x - 4y = -20. \] Divide through by \( -4 \): \[ 3x + y = 5. \]
Conclusion: The relation between \( x \) and \( y \) is \( 3x + y = 5 \).