Why the distance of any point on a ellipse is equal to an equation of a line?

Question

I don't see the geometric reason of the following equation $$\sqrt{(x-c)^2+y^2}=a-\frac{c}{a}x$$ Why the distance of any point $$P(x,y)$$ on a ellipse is equal to an equation of a line? Thank you.

Answer 1

A conics is defined as collection or locus of points $P(x,y)$ which move such that there distance from a fixed point $S(ae,0)$ always equals $e$ times, its perpendicular distance from a fixed line $x=a/e.$ This gives a connection/equation of $x,y$ which represents ellipse if $e<1$, hyperbola is $e>1$ and parabola if $e=1$. $$\sqrt{(x-ae)^2+y^2}=(x-a/e) \implies \frac{x^2}{a^2}+\frac{y^2}{a^2(1-e^2)}=1$$ An equation of a curve is a relationship between $x,y$ which cannot be dissected as line or some thing. For example $x^2+y^2=2x+2y-1$ is equation of circle even if RHS appears to be an equation of a line. Do not dissect and equation like you have thought.