   SEARCH HOME Math Central Quandaries & Queries If the distance from p(1,5) is 3/4 that of the distance of a(4,-3)? find the equation of a locus. Hi Jason,

Let's try a slightly different problem.

Find the locus of a point which moves such that its distance from (1, 0) is 3/5 its distance from (3, -2).

Let (x, y) be on the locus then its distance from (1, 0) is the square root of (x - 1)2 + (y - 0)2 and its distance from (3, -2) is the square root of (x - 3)2 + (y + 2)2. Thus

the square root of (x - 1)2 + (y - 0)2 is 3/5 of the square root of (x - 3)2 + (y + 2)2

Square both sides to get

(x - 1)2 + (y - 0)2 = 9/25 [(x - 3)2 + (y + 2)2]

or

25 [(x - 1)2 + (y - 0)2] = 9 [(x - 3)2 + (y + 2)2]

Expand what is inside the square brackets on each side and simplify. I got

16 x2 + 16 y2 + 4x - 36y - 11 = 0

Since the coefficient of x2 is equal to the coefficient of y2 this is the equation of a circle. If you wish you can put this equation into a more standard form. I got

16 (x + 1/8)2 + 16 (y - 9/8)2 = 622

Now try your problem,
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