Ellipse
Focal Chord — Using $(SP)^2+(S'P)^2-SP\cdot S'P$
nta_pyq_2026_jan
Grade 11
Question:
Let $S$ and $S'$ be the foci of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{9}=1$ and $P(\alpha,\beta)$ be a point on the ellipse in the first quadrant. If $(SP)^2+(S'P)^2-SP\cdot S'P=37$, then $\alpha^2+\beta^2$ is equal to:
Step-by-Step Solution
Key Concept: $a=5$, $b=3$, $c=4$. $SP+S'P=10$. Let $r_1=SP$, $r_2=S'P$. $r_1^2+r_2^2=(r_1+r_2)^2-2r_1r_2=100-2r_1r_2$. Substituting: $100-3r_1r_2=37\Rightarrow r_1r_2=21$.
$r_1r_2=21$, $\alpha=-\tfrac{5}{2}$. $\alpha^2+\beta^2=13$.
Correct Answer: 4