Consider a family of straight line pairs given by $\frac{tx}{a}-\frac{y}{b}+t=0$ and $\frac{x}{a}+\frac{ty}{b}-1=0$ where $t$ is a parameter.
My goal is to show that the set of intersection points of the pairs forms an ellipse.
For solving this, I have multiplied the first equation by $t$ and added it to the other equation, like this:
$$\frac{t^2 x}{a}-\frac{ty}{b}+t^2=0$$ $$\frac{ x}{a}+\frac{ty}{b}-1=0$$
This gives me $$x=a(\frac{1-t^2}{t^2 +1})$$
How do I proceed from here? Particularly, how can I show that these are the coordinates of a point on an ellipse?
Similarly, $$y=b\frac{2t}{1+t^2}$$
$$\text{Use }(a^2-b^2)^2+(2ab)^2=(a^2+b^2)^2$$
$$a=1\implies (1-b^2)^2+(2b)^2=(1+b^2)^2$$
Alternatively, put $t=\tan\theta$ and use double angle formula