What is the value of sin $\theta $ if $\theta $ is the acute angle between the lines whose equations are px + qy = p + q and p(x - y) + q(x + y) = 2q 

The circle $ x ^ 2 + y ^ 2 - 2kx - 2ky + k ^ 2 = 0$ touches the x-axis at P and y-axis at Q. What is PQ equal to?

What is the distance between the foci of the hyperbola $x ^ 2 - 4y ^ 2 = 1$

Let $\vec{p}=\vec{a}-\vec{b},\vec{q}=\vec{a}+\vec{b}$. If $|\vec{a}|=|\vec{b}|=2$ and $\vec{a}\cdot \vec{b}=2$, then what is the value of $|\vec{p}\times \vec{q}|$?