csat-suneung 2022 Q26 (Geometry)

csat-suneung · South-Korea · csat__math 3 marks Conic sections Circle-Conic Interaction with Tangency or Intersection

For an ellipse $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 16 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$, there is a point A in the first quadrant on the ellipse. Among circles that are tangent to both lines $\mathrm { AF } , \mathrm { AF } ^ { \prime }$ and have their center on the y-axis, let C be the circle whose center has a negative y-coordinate. When the center of circle C is B and the area of quadrilateral $\mathrm { AFBF } ^ { \prime }$ is 72, what is the radius of circle C? [3 points]
(1) $\frac { 17 } { 2 }$
(2) 9
(3) $\frac { 19 } { 2 }$
(4) 10
(5) $\frac { 21 } { 2 }$

For an ellipse $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 16 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$, there is a point A in the first quadrant on the ellipse. Among circles that are tangent to both lines $\mathrm { AF } , \mathrm { AF } ^ { \prime }$ and have their center on the y-axis, let C be the circle whose center has a negative y-coordinate. When the center of circle C is B and the area of quadrilateral $\mathrm { AFBF } ^ { \prime }$ is 72, what is the radius of circle C? [3 points]\\
(1) $\frac { 17 } { 2 }$\\
(2) 9\\
(3) $\frac { 19 } { 2 }$\\
(4) 10\\
(5) $\frac { 21 } { 2 }$

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