Question If the distances of P(x, y) from A(5, 1) and B(-1, 5) are equal, then prove that 3x = 2y.
Video Explanation
Explanatory Answer
Distance between two points whose coordinates are (x1, y1) and (x2, y2) = \\sqrt {{({{x}_{2}}-{{x}_{1}})}^{2}+{({{y}_{2}}-{{y}_{1}})}^{2}})
Given information
Distance between P(x, y) and A(5, 1) = distance between P(x, y) and B(-1, 5)
Distance between P and A = \\sqrt {{(x-5)}^{2}+{(y-1)}^{2}})
Distance between P and B = \\sqrt {{(x+1)}^{2}+{(y-5)}^{2}})
Equate the two distances and solve for x and y.
\\sqrt {{(x-5)}^{2}+{(y-1)}^{2}}) = \\sqrt {{(x+1)}^{2}+{(y-5)}^{2}})
Square both the sides of the equation
(x – 5)2 + (y - 1)2 = (x + 1)2 + (y – 5)2
x2 – 10x + 25 + y2 – 2y + 1 = x2 + 2x + 1 + y2 – 10y + 25
-12x = -8y
Or 12x = 8y
Or 3x = 2y