In the coordinate plane, for point $\mathrm { A } ( 0,4 )$ and point P on the ellipse $\frac { x ^ { 2 } } { 5 } + y ^ { 2 } = 1$, let Q be the point other than A among the two points where the line passing through A and P meets the circle $x ^ { 2 } + ( y - 3 ) ^ { 2 } = 1$. When point P passes through all points on the ellipse, what is the length of the figure traced by point Q? [3 points]
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { 2 } { 3 } \pi$
(5) $\frac { 3 } { 4 } \pi$

(12 points)
Let $O$ be the origin of coordinates. Point $M$ is on the ellipse $C: \dfrac{x^2}{2} + y^2 = 1$. The perpendicular from $M$ to the $x$-axis intersects the $x$-axis at $N$. Point $P$ satisfies $\overrightarrow{NP} = \sqrt{2}\,\overrightarrow{NM}$.
(1) Find the trajectory equation of point $P$.
(2) Let point $Q$ be on the line $x = -3$, and $\overrightarrow{OP} \cdot \overrightarrow{PQ} = 1$. Prove that the line $l$ passing through point $P$ and perpendicular to $OQ$ passes through the right focus $F$ of $C$.

In the plane, $A , B$ are two fixed points and $C$ is a moving point. If $\overrightarrow { A C } \cdot \overrightarrow { B C } = 1$, then the locus of point $C$ is
A. a circle
B. an ellipse
C. a parabola
D. a line

Given hyperbola $C : x ^ { 2 } - y ^ { 2 } = m ( m > 0 )$, point $P _ { 1 } ( 5,4 )$ is on $C$, and $k$ is a constant with $0 < k < 1$. Through $P_n$ on the right branch of $C$, draw a line with slope $k$; this line intersects $C$ at another point $Q_n$ on the left branch. The reflection of $Q_n$ across the $y$-axis gives $P_{n+1}$ on the right branch.
(1) Find the coordinates of $P_2$ when $k = \frac{1}{2}$.
(2) Prove that $\{x_n - y_n\}$ is a geometric sequence.
(3) Prove that $S_n = \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1})$ is a constant independent of $n$.

The shape ``$\varnothing$'' can be made into a beautiful ribbon. Consider it as part of the curve $C$ in the figure. It is known that $C$ passes through the origin $O$ , and points on $C$ satisfy: the abscissa is greater than $- 2$ , and the product of the distance to point $F ( 2,0 )$ and the distance to the line $x = a ( a < 0 )$ equals 4 . Then
A. $a = - 2$
B. The point $( 2 \sqrt { 2 } , 0 )$ is on $C$
C. The maximum ordinate of points on $C$ in the first quadrant is 1
D. When point $\left( x _ { 0 } , y _ { 0 } \right)$ is on $C$ , $y _ { 0 } \leqslant \frac { 4 } { x _ { 0 } + 2 }$

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$.
Specify the intersection $Z_A$ of $\mathcal{H}_A$ with the plane with equation $x = (a+d)/2$.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$, and let $Z_A$ be the intersection of $\mathcal{H}_A$ with the plane $x = (a+d)/2$.
If the matrix $A$ has two non-real eigenvalues, how can one see the eigenvalues of $A$ on $\mathcal{H}_A$? (One may consider the intersection of $Z_A$ with the plane with equation $y = 0$.) Can one see a basis of eigenvectors using $\mathcal{H}_A$?

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$.
In the case where $A = \left(\begin{array}{rr} 1 & 7 \\ -1 & 3 \end{array}\right)$ make a perspective drawing illustrating what precedes.

Consider the branch of the rectangular hyperbola $xy = 1$ in the first quadrant. Let $P$ be a fixed point on this curve. The locus of the mid-point of the line segment joining $P$ and an arbitrary point $Q$ on the curve is part of
(a) A hyperbola
(b) A parabola
(c) An ellipse
(d) None of the above.

From a point $P(h,k)$, two tangents are drawn to the parabola $y^2 = 4ax$ which are perpendicular to each other. Find the locus of $P$.

17. The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y ^ { 2 } = 4 a x$ is another parabola with directrix
(A) $\mathrm { x } = - \mathrm { a }$
(B) $x = - a / 2$
(C) $\quad x = 0$
(D) $\quad x = a / 2$

Normals are drawn from the point $P$ with slopes $m _ { 1 } , m _ { 2 } , m _ { 3 }$ to the parabola $y ^ { 2 } = 4 x$. If locus of $P$ with $\mathrm { m } _ { 1 } \mathrm { m} _ { 2 } = \mathrm { a }$ is a part of the parabola itself then finda.

The normal at a point $P$ on the ellipse $x^{2}+4y^{2}=16$ meets the $x$-axis at $Q$. If $M$ is the mid point of the line segment $PQ$, then the locus of $M$ intersects the latus rectums of the given ellipse at the points
(A) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{2}{7}\right)$
(B) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{\sqrt{19}}{4}\right)$
(C) $\left(\pm2\sqrt{3},\pm\frac{1}{7}\right)$
(D) $\left(\pm2\sqrt{3},\pm\frac{4\sqrt{3}}{7}\right)$

The tangent $PT$ and the normal $PN$ to the parabola $y^{2}=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose
(A) vertex is $\left(\frac{2a}{3},0\right)$
(B) directrix is $x=0$
(C) latus rectum is $\frac{2a}{3}$
(D) focus is $(a,0)$

A tangent to the hyperbola $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 2 } = 1$ meets $x$-axis at P and $y$-axis at Q. Lines PR and QR are drawn such that OPRQ is a rectangle (where O is the origin). Then R lies on :
(1) $\frac { 4 } { x ^ { 2 } } + \frac { 2 } { y ^ { 2 } } = 1$
(2) $\frac { 2 } { x ^ { 2 } } - \frac { 4 } { y ^ { 2 } } = 1$
(3) $\frac { 2 } { x ^ { 2 } } + \frac { 4 } { y ^ { 2 } } = 1$
(4) $\frac { 4 } { x ^ { 2 } } - \frac { 2 } { y ^ { 2 } } = 1$

The centres of those circles which touch the circle, $x^2 + y^2 - 8x - 8y - 4 = 0$, externally and also touch the $x$-axis, lie on:
(1) a circle
(2) an ellipse which is not a circle
(3) a hyperbola
(4) a parabola

The locus of the point of intersection of the lines $\sqrt { 2 } x - y + 4 \sqrt { 2 } k = 0$ and $\sqrt { 2 } k x + k y - 4 \sqrt { 2 } = 0$ ( $k$ is any non-zero real parameter) is
(1) an ellipse whose eccentricity is $\frac { 1 } { \sqrt { 3 } }$
(2) a hyperbola whose eccentricity is $\sqrt { 3 }$
(3) a hyperbola with length of its transverse axis $8 \sqrt { 2 }$
(4) an ellipse with length of its major axis $8 \sqrt { 2 }$

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } = 1$ from any of its foci?
(1) $( - 2 , \sqrt { 3 } )$
(2) $( - 1 , \sqrt { 2 } )$
(3) $( - 1 , \sqrt { 3 } )$
(4) $( 1,2 )$

The locus of the mid-point of the line segment joining the focus of the parabola $y ^ { 2 } = 4 a x$ to a moving point of the parabola, is another parabola whose directrix is:
(1) $x = a$
(2) $x = 0$
(3) $x = - \frac { a } { 2 }$
(4) $x = \frac { a } { 2 }$

Let $C$ be the locus of the mirror image of a point on the parabola $y ^ { 2 } = 4x$ with respect to the line $y = x$. Then the equation of tangent to $C$ at $P ( 2,1 )$ is :
(1) $x - y = 1$
(2) $2x + y = 5$
(3) $x + 3y = 5$
(4) $x + 2y = 4$

The locus of the point of intersection of the lines $( \sqrt { 3 } ) k x + k y - 4 \sqrt { 3 } = 0$ and $\sqrt { 3 } x - y - 4 ( \sqrt { 3 } ) k = 0$ is a conic, whose eccentricity is

The locus of the mid-point of the line segment joining the point $( 4,3 )$ and the points on the ellipse $x ^ { 2 } + 2 y ^ { 2 } = 4$ is an ellipse with eccentricity
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 1 } { 2 \sqrt { 2 } }$
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $\frac { 1 } { 2 }$

Let a tangent to the curve $y ^ { 2 } = 24 x$ meet the curve $x y = 2$ at the points $A$ and $B$. Then the midpoints of such line segments $A B$ lie on a parabola with the
(1) directrix $4 x = 3$
(2) directrix $4 x = - 3$
(3) Length of latus rectum $\frac { 3 } { 2 }$
(4) Length of latus rectum 2

The equations of two sides of a variable triangle are $x = 0$ and $y = 3$, and its third side is a tangent to the parabola $y ^ { 2 } = 6 x$. The locus of its circumcentre is:
(1) $4 y ^ { 2 } - 18 y - 3 x - 18 = 0$
(2) $4 y ^ { 2 } + 18 y + 3 x + 18 = 0$
(3) $4 y ^ { 2 } - 18 y + 3 x + 18 = 0$
(4) $4 y ^ { 2 } - 18 y - 3 x + 18 = 0$

Let $P$ be a point on the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$. Let the line passing through $P$ and parallel to $y$-axis meet the circle $x^2 + y^2 = 9$ at point $Q$ such that $P$ and $Q$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $PQ$ such that $PR:RQ = 4:3$ as $P$ moves on the ellipse, is:
(1) $\frac{11}{19}$
(2) $\frac{13}{21}$
(3) $\frac{\sqrt{139}}{23}$
(4) $\frac{\sqrt{13}}{7}$