conic sections
I'm trying to solve this problem but can't understand what is meant by this "polar"
The question is as follows.,
"If the polar of any point with respect to the parabola $y^2=4ax$ touches the circle $y^2+x^2=4a^2$, show that the locus of the point is the curve $x^2-y^2=4a^2$
Thank you.
Animation of the specified locus, for $a = 1/2$:
Without loss of generality, we may assume that the line is the $x$-axis, and that the point is the point $(0,a)$, where $a$ is positive.
A circle with centre $(p,q)$ that touches the $x$-axis must have radius $|q|$. So it has equation $$(x-p)^2+(y-q)^2=q^2.$$ The circle goes through $(0,a)$. It follows that $$(0-p)^2+(a-q)^2=q^2.$$ Simplify. We get $$p^2+a^2-2aq=0.$$ We can rewrite this as $$q=\frac{1}{2a}p^2 +\frac{a}{2},$$ indeed the equation of a parabola.
Best Answer
The term "polar" is standard in the study of conic sections, so it is probably defined in your class notes or textbook. Anyway, the polar of a point $P$ with respect to a conic $C$ is the line that passes through the two points where tangents from $P$ meet $C$.
There is a (pretty poor) description here.
I'll look for a better one.
This page is better, though it only deals with pole/polar of a circle.