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) $24 \sqrt { 3 }$

22. (Total Score: 18 points) Subquestion 1 is worth 6 points, Subquestion 2 is worth 8 points, Subquestion 3 is worth 4 points. Let $P _ { 1 } ( x _ { 1 } , y _ { 1 } )$, $P _ { 2 } ( x _ { 2 } , y _ { 2 } )$, $\cdots$, $P _ { n } ( x _ { n } , y _ { n } )$ ($n \geq 3$, $n \in \mathbb{N}$) be points on a conic section C, and let $a _ { 1 } = \left| OP _ { 1 } \right| ^ { 2 From $\left\{ \begin{array} { l } \frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 25 } = 1 \\ x _ { 3 } ^ { 2 } + y _ { 3 } ^ { 2 } = 70 \end{array} \right.$, we obtain $\left\{ \begin{array} { l } x _ { 3 } ^ { 2 } = 60 \\ y _ { 3 } ^ { 2 } = 10 \end{array} \right.$ $\therefore$ The coordinates of point $P_{3}$ can be $( 2 \sqrt { 15 } , \sqrt { 10 } )$.
(2) Solution 1: The minimum distance from the origin $O$ to each point on the conic curve $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ $(a > b > 0)$ is $b$, and the maximum distance is $a$. $\because a _ { 1 } = \left| O P _ { 1 } \right| ^ { 2 } = a ^ { 2 }$, $\therefore d < 0$, and $a _ { n } = \left| O P _ { n } \right| ^ { 2 } = a ^ { 2 } + ( n - 1 ) d \geq b ^ { 2 }$, $\therefore \frac { b ^ { 2 } - a ^ { 2 } } { n - 1 } \leq d < 0$. $\because n \geq 3$, $\frac { n ( n - 1 ) } { 2 } > 0$, $\therefore S _ { n } = n a ^ { 2 } + \frac { n ( n - 1 ) } { 2 } d$ is increasing on $\left[ \frac { b ^ { 2 } - a ^ { 2 } } { n - 1 } , 0 \right)$, thus the minimum value of $S _ { n }$ is $n a ^ { 2 } + \frac { n ( n - 1 ) } { 2 } \cdot \frac { b ^ { 2 } - a ^ { 2 } } { n - 1 } = \frac { n \left( a ^ { 2 } + b ^ { 2 } \right) } { 2 }$. Solution 2: For each natural number $k$ $(2 \leq k \leq n)$, from $\left\{ \begin{array} { l } x _ { k } ^ { 2 } + y _ { k } ^ { 2 } = a ^ { 2 } + ( k - 1 ) d \\ \frac { x _ { k } ^ { 2 } } { a ^ { 2 } } + \frac { y _ { k } ^ { 2 } } { b ^ { 2 } } = 1 \end{array} \right.$, we solve to get $y _ { k } ^ { 2 } = \frac { - b ^ { 2 } ( k - 1 ) d } { a ^ { 2 } - b ^ { 2 } }$ $\because 0 < y _ { k } ^ { 2 } \leq b ^ { 2 }$, we obtain $\frac { b ^ { 2 } - a ^ { 2 } } { k - 1 } \leq d < 0$, $\therefore \frac { b ^ { 2 } - a ^ { 2 } } { n - 1 } \leq d < 0$. The rest is the same as Solution 1.
(3) Solution 1: If the hyperbola $C: \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, and point $P _ { 1 } ( a , 0 )$, then for a given $n$, the necessary and sufficient condition for the existence of points $P _ { 1 } , P _ { 2 } , \cdots , P _ { n }$ is $d > 0$. $\because$ The distance from the origin $O$ to each point on the hyperbola $C$ is $h \in [ |a| , +\infty )$, and $\left| O P _ { 1 } \right| = a ^ { 2 }$, $\therefore$ Points $P _ { 1 } , P _ { 2 } , \cdots , P _ { n }$ exist if and only if $\left| O P _ { n } \right| ^ { 2 } > \left| O P _ { 1 } \right| ^ { 2 }$, i.e., $d > 0$. Solution 2: If the parabola $C : y ^ { 2 } = 2px$, and point $P _ { 1 } ( 0 , 0 )$, then for a given $n$, the necessary and sufficient condition for the existence of points $P _ { 1 } , P _ { 2 } , \cdots , P _ { n }$ is $d > 0$. The reasoning is the same as above. Solution 3: If the circle $C: ( x - a ) ^ { 2 } + y ^ { 2 } = a ^ { 2 }$ $(a \neq 0)$, and $P _ { 1 } ( 0 , 0 )$, then for a given $n$, the necessary and sufficient condition for the existence of points $P _ { 1 } , P _ { 2 } , \cdots , P _ { n }$ is $0 < d \leq \frac { 4 a ^ { 2 } } { n - 1 }$. $\because$ The minimum distance from the origin $O$ to each point on the circle $C$ is $0$, and the maximum distance is $2 |a|$, and $\left| O P _ { 1 } \right| ^ { 2 } = 0$, $\therefore d > 0$ and $\left| O P _ { n } \right| ^ { 2 } = ( n - 1 ) d \leq 4 a ^ { 2 }$, i.e., $0 < d \leq \frac { 4 a ^ { 2 } } { n - 1 }$.

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) $\sqrt { \frac { a ^ { 2 } b ^ { 2 } + c ^ { 2 } d ^ { 2 } } { c ^ { 2 } + d ^ { 2 } } }$.

6. Let Cl and C 2 be, respectively, the parabolas $\mathrm { x } 2 = \mathrm { y } - 1$ and $\mathrm { y } 2 = \mathrm { a }$. Let P be any point on C 1 and Q be any point on C 2 . Let P 1 and Q 1 be the reflections of P and Q , respectively, with respect to the line y $= \mathrm { x }$. Prove that P 1 lies on $\mathrm { C } 2 , \mathrm { Q } 1$ lies on Cland PQ $\geq \min \{ \mathrm { PP } 1 , \mathrm { QQ } 1 \}$. Hence or otherwise, determine points P0 and Q on the parabolas Cl and C 2 respectively such that $\mathrm { P } 0 \mathrm { Q } \leq \mathrm { P } 0$ for all pairs of points $( \mathrm { P } , \mathrm { Q } )$ with P on C 1 and Q on C . 7.
(a)
$$\begin{aligned} & \text { Suppose } P ( x ) = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \ldots \ldots . . + a _ { n } x ^ { 7 } . \text { If } | p ( x ) | \leq \left| e ^ { x - 1 } - 1 \right| \text { for all } \\ & x \geq 0 \text {, prove that } \left| a _ { 1 } + 2 a _ { 2 } + \ldots \ldots . + n a _ { n } \right| \leq 1 \end{aligned}$$
(b) For $x > 0$, let $f ( x ) = \int _ { 1 } ^ { x } \frac { \ln t } { 1 + \tau } d t$. Find the function $f ( x ) + f ( 1 / x )$ and show that $f ( e ) + f ( 1 / e ) = 1 / 2$. Here $\ln t = \log _ { e } t$.

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) $\frac { \pi } { 3 }$