Can you please help me with this math problem?

Question

Let L be a line in the plane, F be a point in the same plane outside of the line L. Let X be any point in the same plane, dL be the distance from X to L (so it is a perpendicular line). df be the distance from X to F. Let us consider the locus of all points X that df/dl = e where e is a constant positive number.

If e = 1, we have our standard definition of a parabola. The remarkable fact is that if 01, the locus is a hyperbola. So, we have the uniform definition of all three conics depending only on the value of eccentricity e.

Can you start with a star (*) above the line L , express it in coordinates and by a series of algebraic equations, get our standard equations of an ellipse and hyperbola (may be – shifted) depending on the value of e (01).

Can you please write a complete proof in general form and for the coordinates of the point P and equation of line L you can use specific values which are convenient for you.

Answer 1

I answered this before
Let´s take L as x=0 and F(c,0) and e=c/a whera is not defined yet ( c/a is called excentricity)
If (x,y ) is a point of the locus
[(x-c)^2+y^2]/x^2 =(c/a)^2 which gives you(shifted at the x axis)
you the equations.

Answer 2

2=325