Question

A curve is traced by a point p(x,y) which moves such that its distance from the point a(-1,1) is three times its distance from the point b(2,-1). determine the equation of the curve.

Answer 1

Refer to the figure shown below.

The point c, on line segment ab, is in a 3:1 ratio from a.
Therefore,
the x-coordinate of c is -1 +(3/4)*(3) = 5/4 = 1.25.
the y-coordinate of c is 1 - (3/4)*(2) = -1/2 = -0.5.

At point p(x,y), we want d1 = 3d₂, or d₁² = 9d₂².
Therefore
(x + 1)² + (y - 1)² = 9[(x - 2)² + (y + 1)²]
x² + 2x + 1 + y² - 2y + 1 = 9x² - 36x + 36 + 9y² + 18y + 9
8x² - 38x + 8y² + 20y + 43 = 0
x² - 4.75x + y² + 2.5y + 5.375 = 0
(x - 2.375)² - 5.6406 + (y + 1.25)² - 1.5625 + 5.375 = 0
(x - 2.375)² + (y + 1.25)² = 1.828
This is a circle with center at (2.375, -1.25) and radius 1.352.

Answer:
(x - 2.375)² + (y +1.25)² = 1.828