Basic Locus Problem
Geometry
Level
4
P moves in a plane such that PA=λPB,where A,B are fixed points and λ>0.Which of the following can be locii for P?
parabola
ellipse
hyperbola
circle,straight lines
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1 solution
Let P(x,y) be any point in the xy-plane, and let A(-c/2,0), B(c/2,0) be two fixed points on the x-axis. We are told |PA| = k*|PB| (k > 0). This can be expressed as:
sqrt[(x + c/2)^2 + y^2] = k*sqrt[(x - c/2)^2 + y^2];
or (x + c/2)^2 + y^2 = k^2 * [(x - c/2)^2 + y^2];
or x^2 +cx + c^2/4 + y^2 = k^2 *(x^2 - cx + c^2/4 + y^2);
or 0 = (k^2 - 1) x^2 - c (k^2 + 1) x + (k^2 - 1) y^2 + (k^2 - 1)(c^2/4);
or 0 = x^2 - c [(k^2 + 1)/(k^2 - 1)] x + y^2 + c^2 /4;
or (c^2/4) [(k^2 + 1)/(k^2 - 1)]^2 - c^2/4 = {x^2 - c [(k^2 + 1)/(k^2 - 1)] x + (c^2/4) [(k^2 + 1)/(k^2 - 1)]^2} + y^2;
or k [c/(k^2 - 1)]^2 = [x - (c/2) [(k^2 + 1)/(k^2 - 1)]]^2 + y^2.
We ultimately end up with a locus that resembles (x - a)^2 + (y - b)^2 = r^2, or a circle.