Expanding this and using the fact that the insersection must satisfy both $f_1 = 0$ and $f_2 = 0$, we obtain the requisite ewsult, that $2aa' + 2bb' = c + c'$.

Two circles have equations (i) $x^2 + y^2 + 2ax + 2by + c = 0$ and (ii) $x^2 + y^2 + 2a'x + 2b'y + c' = 0$. Show that these circles are orthogonal for $2aa' + 2bb' = c + c'$.

Spoiler

a) A mass distribution in the positive $x$ region of the $xy$-plane and in the shape of a semicircle of radius $a$, centered on the origin, has mass per unit area $k$. Find using plane polar coordinates:

- Its mass $M$
- The coordinates $(\overline{x}, \overline{y})$ of its center of mass
- Its moments of inertia about the $x$ and $y$ axes

Spoiler

b) Do as above for a semi-infinite sheet with mass per unit area \[ \rho = ke^{-(x^2 + y^2)/a^2} \]

for $x \geq 0$ and $\rho = 0$ for $x < 0$, and where $a$ is constant.

Spoiler

c) Evaluate the following integral: \[ \int_0^a \int_0^{\sqrt{a^2 - y^2}} (x^2 + y^2) \tan^{-1}(y/x) \, \de{x} \, \de{y}. \]

Spoiler