Match the following:
P. The normal at an end of a latusrectum of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ passes through an end of the minor axis if $e^4$ is equal to
Q. PQ is a double ordinate of a parabola $y^2 = 4ax$. If the locus of its point of trisection is another parabola length of whose latus rectum is k times the length of the latus rectum of the given parabola then k is equal to
R. If e and $e'$ are the distances of the extremities of any focal chord from the focus f of the parabola $y^2 = 4ax$, then $\frac{1}{e} + \frac{1}{e'}$ is equal to
S. If e and $e'$ be the eccentricities of a hyperbola and its conjugate, then $\frac{1}{e} + \frac{1}{e'^2}$ is equal to
Step-by-Step Solution
Key Concept: Normal lines to conic sections have specific geometric properties; for an ellipse, the normal at a latus rectum point imposes a constraint that relates eccentricity to the parameter $b$.
(P) For a latus rectum endpoint $(ae, b\sqrt{1-e^2})$ of an ellipse, the normal equation is $\frac{x-ae}{ae/e^2} = \frac{y-b\sqrt{1-e^2}}{-b\sqrt{1-e^2}/b^2}$. For this normal to pass through $(0, -b)$, we derive the condition $e^4 + e^2 = 1$. (Q) For a parabola $y^2 = 4ax$ with double ordinate at $PQ$, the intersection points $R$ and $S$ with focal chord satisfy $PR = RS = SQ$ when the parameters satisfy the given relationship.
Correct Answer: 1