csat-suneung 2022 Q28 (Geometry)

csat-suneung · South-Korea · csat__math 4 marks Conic sections Focal Distance and Point-on-Conic Metric Computation

For two positive numbers $a , p$, let $\mathrm { F } _ { 1 }$ be the focus of the parabola $( y - a ) ^ { 2 } = 4 p x$, and let $\mathrm { F } _ { 2 }$ be the focus of the parabola $y ^ { 2 } = - 4 x$.
When segment $\mathrm { F } _ { 1 } \mathrm {~F} _ { 2 }$ meets the two parabolas at points $\mathrm { P } , \mathrm { Q }$ respectively, $\overline { \mathrm { F } _ { 1 } \mathrm {~F} _ { 2 } } = 3$ and $\overline { \mathrm { PQ } } = 1$. What is the value of $a ^ { 2 } + p ^ { 2 }$? [4 points]
(1) 6
(2) $\frac { 25 } { 4 }$
(3) $\frac { 13 } { 2 }$
(4) $\frac { 27 } { 4 }$
(5) 7

For two positive numbers $a , p$, let $\mathrm { F } _ { 1 }$ be the focus of the parabola $( y - a ) ^ { 2 } = 4 p x$, and let $\mathrm { F } _ { 2 }$ be the focus of the parabola $y ^ { 2 } = - 4 x$.

When segment $\mathrm { F } _ { 1 } \mathrm {~F} _ { 2 }$ meets the two parabolas at points $\mathrm { P } , \mathrm { Q }$ respectively, $\overline { \mathrm { F } _ { 1 } \mathrm {~F} _ { 2 } } = 3$ and $\overline { \mathrm { PQ } } = 1$. What is the value of $a ^ { 2 } + p ^ { 2 }$? [4 points]\\
(1) 6\\
(2) $\frac { 25 } { 4 }$\\
(3) $\frac { 13 } { 2 }$\\
(4) $\frac { 27 } { 4 }$\\
(5) 7

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