Express triangle sides given a circle and an angle. Then, find its geometric locus.

Question

Given the equation of the circle:

$$x^2 + (y-a)^2 = a^2$$

• How to express $ON$ and $PN$ in terms of $a$ and $\phi$ ?

• Hence, how to find the locus of $P$?

Answer 1

Use the right triangles △OAN and △ONP to derive ON and PN, respectively,

$$ ON = 2a \tan \phi$$ $$ PN = ON \tan \phi = 2a \tan^2 \phi$$

To find the locus of P, write the function of P in its polar coordinates, with the help of right triangle △ONP,

$$ON^2+PN^2 = 4a^2 (\tan^2 \phi + \tan^4 \phi)=r^2$$

After simplification,

$$ r = 2a \frac{\sin\phi}{\cos^2\phi} $$

Then, it is more convenient to view above curve in its $xy$-coordinates with

$$x=r\cos\phi,\space y=r\sin\phi$$

which is just,

$$ x^2 = 2ay$$

Hence, P is a parabola.