I was reading a book about polar coordinates. There was a section about the "Equation of a chord (joining $\theta=\alpha-\beta $ and $\theta=\alpha+\beta$) in a conic section".The proof given in the book is as follows:
Let $\frac{l}{r}=1-ecos\theta$ and $\frac{l}{r}=acos\theta +bsin\theta$ be the equations of the conic and the chord respectively(, where $e$ is the eccentricity and $l$ is the semi latus rectum of the conic). For the common points of them, $1-ecos\theta=acos\theta +bsin\theta$. At the common points $\theta=\alpha-\beta $ and $\theta=\alpha+\beta$, we have, $1-ecos(\alpha+\beta)=acos(\alpha+\beta )+bsin(\alpha+\beta)$ and $1-ecos(\alpha - \beta)=acos(\alpha-\beta )+bsin(\alpha-\beta)$. From these two, $(a+e)cos(\alpha-\beta)+bsin(\alpha-\beta)-1=0$ and $(a+e)cos(\alpha +\beta)+bsin(\alpha +\beta)-1=0$. By cross multiplication, $\frac{a+e}{-sin(\alpha-\beta)+sin(\alpha+\beta)}=\frac{b}{-cos(\alpha+\beta)+cos(\alpha-\beta)}=\frac{1}{cos(\alpha-\beta)sin(\alpha+\beta)-sin(\alpha-\beta)cos(\alpha+\beta)}$. Hence, $a+e=\frac{cos\alpha}{cos\beta}$ or $a=\frac{cos\alpha}{cos\beta}-e$ and $b=\frac{sin\alpha}{cos\beta}$. Thus , the equation of the chord is $\frac{l}{r}=(\frac{cos\alpha}{cos\beta}-e)cos\theta + \frac{sin\alpha}{cos\beta}sin\theta$ or, $\frac{l}{r}=-ecos\theta+sec\beta cos(\theta-\alpha)$.
However, if $\theta=\alpha-\beta $ and $\theta=\alpha+\beta$, then $\alpha=\theta$ and $\beta=0$. Hence,$\frac{l}{r}=-ecos\theta+sec\beta cos(\theta-\alpha)=-ecos\theta$. Thus, $\frac{l}{r}=-ecos\theta$ , which is the equation of the directrix if $(r,\theta)$ is a point on the directrix . However, this doesn't make any sense?So, why are they assuming $\theta=\alpha-\beta $ and $\theta=\alpha+\beta$ as the common point ? Also, is the proof given in the book , valid?So, is the equation of the chord which meets the conic section at $(r,\theta)$ is $\frac{l}{r}=-ecos\theta$ ? Also , does the title make any sense? I mean the common chord "joining the points $\theta=\alpha-\beta $ and $\theta=\alpha+\beta$" , but $\theta$ is a single point, so what does they mean ? I am not quite getting it...