If a = b cos(2π/3) = c cos(4π/3), Find ab + bc + ca
Question:
If
\[
a=b\cos\frac{2\pi}{3}=c\cos\frac{4\pi}{3}
\]
find
\[
ab+bc+ca
\]
Solution
\[ \cos\frac{2\pi}{3}=-\frac12, \qquad \cos\frac{4\pi}{3}=-\frac12 \]
Given, \[ a=-\frac{b}{2}=-\frac{c}{2} \]
\[ b=-2a, \qquad c=-2a \]
Now, \[ ab+bc+ca \]
\[ = a(-2a)+(-2a)(-2a)+a(-2a) \]
\[ = -2a^2+4a^2-2a^2 \]
\[ \boxed{0} \]