If a variable tangent of the circle  $x^2+y^2=1$ intersect the ellipse $x^2+2y^2=4$ at P and Q then the locus of the points of intersection of the tangents are P and Q is

$x^2+y^2=1,x^2+2y^2=4$

$R\left(x_1,y_1\right)$ is point of intersection of tangents drawn at P, Q to ellipse PQ chord equation is $x{x}_1+2y{y}_1-4=0$ touches $x^2+y^2=1\Rightarrow 1^2\left[x_1^2+{\left(2y_1\right)}^2\right]=16\Rightarrow x^2+4y^2=16$

is ellipse with $e=\sqrt{\frac{16-4}{16}}=\frac{\sqrt{3}}{2}$ 

Latus rectum length $=\frac{2\left(4\right)}{4}=2$