a point F is at a certain distance from a line AB .Another point P is moving in a plane such that the ratio of it's distance from F and line AB remains constant .find the locus of the point P

Question

a point F is at a certain distance from a line AB  .Another point  P is moving in a plane such that the ratio of it's distance from F and line AB remains constant .find the locus of the point P

Steve · Answer

we can without loss of generality let the line be the x-axis, and F be at (0,k+1). We know that if the ratio is 1, the locus is a parabola. Maybe it still is. If P starts at (0,1) the ratio is k. We have y = √(x^2 + (k-y)^2) y^2 = x^2 + y^2 - 2ky + k^2 2ky = x^2+k^2 it's still a parabola.