Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics IV / 2

For a fixed point $F$ not on a fixed line $D$, the ellipse with focus of a conic $F$, directrix of a conic $D$ and eccentricity $0<e<1$ is the locus of points $P$ satisfyingSet $c=\sqrt{a^2-b^2}$ and $e=c/a$. One matched focus of a conic and directrix of a conic pair isIndeed the squared distance condition isExpanding and cancelling the terms linear in $x$ giveswhich is exactly $x^2/a^2+y^2/b^2=1$. Both sides of the unsquared distance condition are nonnegative, so squaring introduces no extra points. Reflecting across the $y$-axis gives the other matched pair: $F=(-c,0)$ and $D:x=-a^2/c$, with the same eccentricity.

Use the point ${\mathbf{p}}_0=(-2,-2,4)$ and the orthonormal basis of directions in the cutting planeBoth vectors are orthogonal to the normal vector $(1,1,2)$, so every point of the plane has unique coordinatesSubstituting into the cone equation givesConsequently the intersection has equationBecause these are orthonormal coordinates, their lengths are the actual lengths in the plane. Thus this is an ellipse with semi-major axis $\sqrt{24}$ and semi-minor axis $\sqrt{8}$, not merely a projected ellipse. By a, its eccentricity is