The following statement is given in my book under the topic Tangents to an Ellipse:
The eccentric angles of the points of contact of two parallel tangents differ by $\pi$
In case of a circle, it is easy for me to visualise that two parallel tangents meet the circle at two points which are apart by $\pi$ radians as they are diametrically opposite. But in case of ellipse, as the eccentric angle is defined with respect to the auxiliary circle and not the ellipse, I am unable to understand why two parallel tangents meet the ellipse at points which differ by $\pi$.
Kindly explain the reason behind this fact.
Best Answer
The reason is that an ellipse can be obtained by stretching/shrinking a circle. The strech/shrink is a linear map (linear transformation).
Let's consider two tangent lines on the circle $x^2+y^2=a^2$ at $(a\cos\theta,a\sin\theta)$,$(a\cos(\theta+\pi),a\sin(\theta+\pi))$. You already know that the two tangent lines are parallel.
Now, let's stretch/shrink the circle and the tangent lines. Stretching/shrinking the circle $x^2+y^2=a^2$ to obtain the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ means that you replace $y$ in $x^2+y^2=a^2$ with $\frac{a}{b}y$ to have $x^2+\left(\frac aby\right)^2=a^2$ which is nothing but $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
By this stretch/shrink, we have the followings :
The circle $x^2+y^2=a^2$ is transformed to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
The two parallel lines are transformed to two parallel lines.
The two lines tangent to the cirlce are transformed to two lines tangent to the ellipse.
The tangent points $(a\cos\theta,a\sin\theta)$,$(a\cos(\theta+\pi),a\sin(\theta+\pi))$ on the circle are transformed to two tangent points $(a\cos\theta,b\sin\theta)$,$(a\cos(\theta+\pi),b\sin(\theta+\pi))$ on the ellipse respectively.
From the above facts, it follows that the eccentric angles of the points of contact of two parallel tangents differ by $\pi$.
The followings are the proof for the above facts.
Let's consider the circle $x^2+y^2=a^2$ and two points $(a\cos\theta,a\sin\theta)$,$(a\cos(\theta+\pi),a\sin(\theta+\pi))$.
The equation of the tangent line at $(a\cos\theta,a\sin\theta)$ is given by $$a\cos\theta\ x+a\sin\theta\ y=a^2\tag1$$
Similarly, the equation of the tangent line at $(a\cos(\theta+\pi),a\sin(\theta+\pi))$ is given by $$a\cos(\theta+\pi)x+a\sin(\theta+\pi)y=a^2\tag2$$
Now, let's stretch/shrink the circle and the lines $(1)(2)$ by replacing $y$ with $\frac aby$ to have $$(1)\to a\cos\theta\ x+a\sin\theta\cdot\frac aby=a^2\tag3$$ $$(2)\to a\cos(\theta+\pi)x+a\sin(\theta+\pi)\cdot\frac aby=a^2\tag4 $$ Here note that these lines $(3)(4)$ are parallel since the slope of each line is $\frac{-b\cos\theta}{a\sin\theta}$.
Finally, note that $(3)$ can be written as $$\frac{a\cos\theta}{a^2}x+\frac{b\sin\theta}{b^2}y=1\tag5$$ which is nothing but the tangent line at $(a\cos\theta,b\sin\theta)$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
Similarly, $(4)$ can be written as $$\frac{a\cos(\theta+\pi)}{a^2}x+\frac{b\sin(\theta+\pi)}{b^2}y=1\tag6$$ which is nothing but the tangent line at $(a\cos(\theta+\pi),b\sin(\theta+\pi))$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
Since $(5)(6)$ are parallel, we see that the eccentric angles of the points of contact of two parallel tangents differ by $\pi$. $\quad\square$