In coordinate space, there is a point $\mathrm { A } ( 9,0,5 )$, and on the $xy$-plane there is an ellipse $\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 1$. For a point P on the ellipse, find the maximum value of $\overline { \mathrm { AP } }$. [3 points]

There is an ellipse $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$. For a point P on the circle $x ^ { 2 } + ( y - 3 ) ^ { 2 } = 4$, let Q be the point with positive $y$-coordinate among the points where the line $\mathrm { F } ^ { \prime } \mathrm { P }$ meets this ellipse. Find the maximum value of $\overline { \mathrm { PQ } } + \overline { \mathrm { FQ } }$. [4 points]

In a plane, there is an equilateral triangle ABC with side length 10. For a point P satisfying $\overline { \mathrm { PB } } - \overline { \mathrm { PC } } = 2$, when the length of segment PA is minimized, what is the area of triangle PBC? [4 points]
(1) $20 \sqrt { 3 }$
(2) $21 \sqrt { 3 }$
(3) $22 \sqrt { 3 }$
(4) $23 \sqrt { 3 }$
(5) $24 \sqrt { 3 }$

12. In the rectangular coordinate system $x O y$, let $P$ be a moving point on the right branch of the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$. If the distance from point $P$ to the line $x - y + 1 = 0$ is always greater than or equal to c, then the maximum value of the real number c is $\_\_\_\_$.

Let the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ have eccentricity $\frac{2\sqrt{2}}{3}$, with lower vertex $A$ and right vertex $B$, $|AB| = \sqrt{10}$.
(1) Find the standard equation of the ellipse.
(2) Let $P$ be a moving point not on the $y$-axis, and let $R$ be a point on the ray $AP$ satisfying $|AR| \cdot |AP| = 3$.
(i) If $P(m, n)$, find the coordinates of point $R$ (expressed in terms of $m, n$).
(ii) Let $O$ be the origin, and $Q$ be a moving point on $C$. The slope of line $OR$ is 3 times the slope of line $OP$. Find the maximum value of $|PQ|$.

(17 points) Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $\frac{2\sqrt{2}}{3}$, lower vertex $A$, right vertex $B$, and $|AB| = \sqrt{10}$.
(1) Find the equation of $C$.
(2) Let $P$ be a moving point not on the $y$-axis, and let $R$ be a point on the ray $AP$ satisfying $|AP| \cdot |AR| = 3$.
(i) If $P(m, n)$, find the coordinates of $R$ (expressed in terms of $m, n$).
(ii) Let $O$ be the origin, and $Q$ be a moving point on $C$. The slope of line $OR$ is 3 times the slope of line $OP$. Find the maximum value of $|PQ|$.

Let $a , b , c , d > 0$, be any real numbers. Then the maximum possible value of $c x + d y$, over all points on the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, must be
(A) $\sqrt { a ^ { 2 } c ^ { 2 } + b ^ { 2 } d ^ { 2 } }$.
(B) $\sqrt { a ^ { 2 } b ^ { 2 } + c ^ { 2 } d ^ { 2 } }$.
(C) $\sqrt { \frac { a ^ { 2 } c ^ { 2 } + b ^ { 2 } d ^ { 2 } } { a ^ { 2 } + b ^ { 2 } } }$.
(D) $\sqrt { \frac { a ^ { 2 } b ^ { 2 } + c ^ { 2 } d ^ { 2 } } { c ^ { 2 } + d ^ { 2 } } }$.

If the tangent at a point on the ellipse $\frac{x^2}{27} + \frac{y^2}{3} = 1$ meets the coordinate axes at $A$ and $B$, and $O$ is the origin, then the minimum area (in sq. units) of the triangle $OAB$ is:
(1) $\frac{9}{2}$
(2) $9$
(3) $9\sqrt{3}$
(4) $\frac{\sqrt{3}}{2}$

Let $P$ be a point on the parabola $x ^ { 2 } = 4 y$. If the distance of $P$ from the center of the circle $x ^ { 2 } + y ^ { 2 } + 6 x + 8 = 0$ is minimum, then the equation of the tangent to the parabola at $P$ is
(1) $x + y + 1 = 0$
(2) $x + 4 y - 2 = 0$
(3) $x + 2 y = 0$
(4) $x - y + 3 = 0$

If the point $P$ on the curve, $4 x ^ { 2 } + 5 y ^ { 2 } = 20$ is farthest from the point $Q ( 0 , - 4 )$, then $PQ ^ { 2 }$ is equal to
(1) 36
(2) 48
(3) 21
(4) 29

Let a tangent be drawn to the ellipse $\frac { x ^ { 2 } } { 27 } + y ^ { 2 } = 1$ at $( 3 \sqrt { 3 } \cos \theta , \sin \theta )$ where $\theta \in \left( 0 , \frac { \pi } { 2 } \right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to:
(1) $\frac { \pi } { 8 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 6 }$
(4) $\frac { \pi } { 3 }$