p²+q²/p²cosθ+q²sinθ
(p²+q²)sinθ/(pcosθ+qsinθ)²
p²+q²cosθ/pcosθ+qsinθ
(p²+q²)sinθ/pcosθ+qsinθ
In the triangle BCD
cosα = q/√p²+q² and sinα = p/√p²+q²
Using sine rule in triangle ABD
AB/sinθ = BD/sin(θ+α) ⇒ AB = √p²+q² sinθ/sinθcosα + cosθsinα = √p²+q² sinθ/sinθ⋅q/√p²+q² + cosθ⋅p/√p²+q² ⇒ AB = (p²+q²)sinθ/(pcosθ+qsinθ)