Derivation of general parametric equation of chord on ellipse

Question

How is the equation of a chord for the ellipse in parametric form given 2 points $P(a\cos\theta,b\sin\theta)$ and $Q(a\cos\varphi,b\sin\varphi)$ as $$bx\cos((\theta+\varphi)/2) + ay\sin((\theta+\varphi)/2) = ab\cos((\theta-\varphi)/2)$$ derived? I am not sure how to proceed after using point-gradient form. Any help is greatly appreciated, thanks.

Answer 1

Recall the two-point form of line $$ \frac{y-y_P}{x-x_P}=\frac{y_Q-y_P}{x_Q-x_P} $$ yields $$ \frac{y-b\sin\theta}{x-a\cos\theta}=\frac{b\sin\varphi-b\sin\theta}{a\cos\varphi-a\cos\theta}. $$ Clearing denominators, $$ (y-b\sin\theta)(a\cos\varphi-a\cos\theta)=(b\sin\varphi-b\sin\theta)(x-a\cos\theta) $$ and use sum-to-product formulae, $$ -a\sin\frac{\varphi-\theta}2\sin\frac{\varphi+\theta}2(y-b\sin\theta)=b\cos\frac{\varphi+\theta}2\sin\frac{\varphi-\theta}2(x-a\cos\theta) $$ Removing the common factor $\sin\frac{\varphi-\theta}2$ and rearranging, \begin{align*} b\cos\frac{\varphi+\theta}2x+a\sin\frac{\varphi+\theta}2y&=ab\left[\cos\theta\cos\frac{\varphi+\theta}2+\sin\frac{\varphi+\theta}2\sin\theta\right]\\ &=ab\cos\left(\theta-\frac{\varphi+\theta}2\right)\\ &=ab\cos\frac{\theta-\varphi}2 \end{align*}