Chord bisected by parabola; exhaustive range; counting
MMTS_Full_Test_10
Grade 12
Question:
Three distinct chords of $x^2+4y^2=2000$ from $P(0,a)$ are bisected by $x^2=20y$. Exhaustive set of $a$ is $(k_1,k_2)$. Number of positive integral solutions of $x+y=k_2-k_1$ is
(A) 169
(B) 19
(C) 51
(D) 969
Step-by-Step Solution
Key Concept: Chord of $x^2+4y^2=2000$ bisected at $(10t,5t^2)$: equation passes through $P(0,a)$ iff $a=5t^2+5$. Distinct chords require $0<t^2<4$. So $5<a<25$: $(k_1,k_2)=(5,25)$.
19 solutions.
Correct Answer:(B) 19
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