intersection of ellipse with a line parallel to the foci

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consider an ellipse with foci $(x_a,y_a)$ and $(x_b,y_b)$ such that $\sqrt{(x-x_a)^2+(y-y_a)^2}+\sqrt{(x-x_b)^2+(y-y_b)^2}=p$

consider a line parallel to the line through their foci, e.g. $y=\frac{y_b-y_a}{x_b-x_a}x+q$

find the points of intersection

is there a quick way to do this by converting the equation for the ellipse into a conic equation?

cheers, dave xx