Given the ellipse $C : x ^ { 2 } + 3 y ^ { 2 } = 3$, a line passing through point $D ( 1,0 )$ but not through point $E ( 2,1 )$ intersects the ellipse $C$ at points $A$ and $B$. The line $AE$ intersects the line $x = 3$ at point $M$.\n(1) Find the eccentricity of the ellipse $C$;\n(II) If $AB$ is perpendicular to the $x$-axis, find the slope of line $BM$;\n(III) Determine the positional relationship between line $BM$ and line $DE$, and explain the reason.

18. The ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ ($a > b > 0$) passes through the point $(0, \sqrt { 2 })$, and has eccentricity [Figure]
(1) Find the equation of ellipse $E$;
(2) The line $x = m y - 1$ ($m \in \mathbb{R}$) intersects the ellipse $E$ at points $A$ and $B$. Determine the positional relationship between the point $G \left( - \frac { 9 } { 4 } , 0 \right)$ and the circle with diameter $AB$, and explain the reason.

(1) Find the equation of ellipse $E$;
(1) Find the equation of ellipse $E$;

(2) The line $x = m y - 1$ ($m \in \mathbb{R}$) intersects the ellipse $E$ at points $A$ and $B$. Determine the positional relationship between the point $G \left( - \frac { 9 } { 4 } , 0 \right)$ and the circle with diameter $AB$, and explain the reason. [Figure]

A circle with the vertex of parabola $C$ as its center intersects $C$ at points $A , B$, and intersects the directrix of $C$ at points $D , E$. If $|DE| = 2 \sqrt { 5 }$, then the distance from the focus of $C$ to the directrix is
(A) 2
(B) 4
(C) 6
(D) 8

Let $a$ be a real number. The number of distinct solutions $(x, y)$ of the system of equations $(x - a)^2 + y^2 = 1$ and $x^2 = y^2$, can only be
(A) $0, 1, 2, 3, 4$ or 5
(B) 0, 1 or 3
(C) $0, 1, 2$ or 4
(D) $0, 2, 3$, or 4

Let $a$ be a real number. The number of distinct solutions $( x , y )$ of the system of equations $( x - a ) ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } = y ^ { 2 }$, can only be
(a) $0,1,2,3,4$ or 5
(b) 0, 1 or 3
(c) 0, 1, 2 or 4
(d) $0,2,3$, or 4

If a circle intersects the hyperbola $y = 1 / x$ at four distinct points $\left( x _ { i } , y _ { i } \right) , i = 1,2,3,4$, then prove that $x _ { 1 } x _ { 2 } = y _ { 3 } y _ { 4 }$.

Let $a$ be a real number. The number of distinct solutions $( x , y )$ of the system of equations $( x - a ) ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } = y ^ { 2 }$, can only be
(a) $0,1,2,3,4$ or 5
(b) 0, 1 or 3
(c) 0, 1, 2 or 4
(d) $0,2,3$, or 4

If a circle intersects the hyperbola $y = 1 / x$ at four distinct points $\left( x _ { i } , y _ { i } \right) , i = 1,2,3,4$, then prove that $x _ { 1 } x _ { 2 } = y _ { 3 } y _ { 4 }$.

For real numbers $a , b , c , d , a ^ { \prime } , b ^ { \prime } , c ^ { \prime } , d ^ { \prime }$, consider the system of equations $$\begin{aligned} a x ^ { 2 } + a y ^ { 2 } + b x + c y + d & = 0 \\ a ^ { \prime } x ^ { 2 } + a ^ { \prime } y ^ { 2 } + b ^ { \prime } x + c ^ { \prime } y + d ^ { \prime } & = 0 \end{aligned}$$ If $S$ denotes the set of all real solutions $( x , y )$ of the above system of equations, then the number of elements in $S$ can never be
(A) 0.
(B) 1.
(C) 2.
(D) 3.

Consider the two curves $$\begin{aligned} & C _ { 1 } : y ^ { 2 } = 4 x \\ & C _ { 2 } : x ^ { 2 } + y ^ { 2 } - 6 x + 1 = 0 \end{aligned}$$ Then,
(A) $C _ { 1 }$ and $C _ { 2 }$ touch each other only at one point
(B) $C _ { 1 }$ and $C _ { 2 }$ touch each other exactly at two points
(C) $C _ { 1 }$ and $C _ { 2 }$ intersect (but do not touch) at exactly two points
(D) $C _ { 1 }$ and $C _ { 2 }$ neither intersect nor touch each other

Let the circles $C_1 : x^2 + y^2 = 9$ and $C_2 : (x-3)^2 + (y-4)^2 = 16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3 : (x-h)^2 + (y-k)^2 = r^2$ satisfies the following conditions: (i) centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$, (ii) $C_1$ and $C_2$ both lie inside $C_3$, and (iii) $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$.
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2 = 8\alpha y$.
List-I: (I) $2h + k$ (II) $\frac{\text{Length of } ZW}{\text{Length of } XY}$ (III) $\frac{\text{Area of triangle } MZN}{\text{Area of triangle } ZMW}$ (IV) $\alpha$
List-II: (P) $6$ (Q) $\sqrt{6}$ (R) $\frac{5}{4}$ (S) $\frac{21}{5}$ (T) $2\sqrt{6}$ (U) $\frac{10}{3}$
Which of the following is the only CORRECT combination?
(A) (I), (S)
(B) (I), (U)
(C) (II), (Q)
(D) (II), (T)

Let the circles $C_1 : x^2 + y^2 = 9$ and $C_2 : (x-3)^2 + (y-4)^2 = 16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3 : (x-h)^2 + (y-k)^2 = r^2$ satisfies the following conditions: (i) centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$, (ii) $C_1$ and $C_2$ both lie inside $C_3$, and (iii) $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$.
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2 = 8\alpha y$.
List-I: (I) $2h + k$ (II) $\frac{\text{Length of } ZW}{\text{Length of } XY}$ (III) $\frac{\text{Area of triangle } MZN}{\text{Area of triangle } ZMW}$ (IV) $\alpha$
List-II: (P) $6$ (Q) $\sqrt{6}$ (R) $\frac{5}{4}$ (S) $\frac{21}{5}$ (T) $2\sqrt{6}$ (U) $\frac{10}{3}$
Which of the following is the only INCORRECT combination?
(A) (I), (P)
(B) (IV), (U)
(C) (III), (R)
(D) (IV), (S)

If the curves $y ^ { 2 } = 6 x , 9 x ^ { 2 } + b y ^ { 2 } = 16$ intersect each other at right angles, then the value of $b$ is:
(1) $\frac { 9 } { 2 }$
(2) 6
(3) $\frac { 7 } { 2 }$
(4) 4

If the circles $x^2 + y^2 - 16x - 20y + 164 = r^2$ and $(x-4)^2 + (y-7)^2 = 36$ intersect at two distinct points, then:
(1) $r > 11$
(2) $0 < r < 1$
(3) $1 < r < 11$
(4) $r = 11$

If the parabolas $y ^ { 2 } = 4 b ( x - c )$ and $y ^ { 2 } = 8 a x$ have a common normal, then which one of the following is a valid choice for the ordered triad $( a , b , c )$
(1) $( 1,1,3 )$
(2) $\left( \frac { 1 } { 2 } , 2,0 \right)$
(3) $\left( \frac { 1 } { 2 } , 2,3 \right)$
(4) All of above

The tangent to the parabola $y ^ { 2 } = 4 x$ at the point where it intersects the circle $x ^ { 2 } + y ^ { 2 } = 5$ in the first quadrant, passes through the point:
(1) $\left( \frac { 1 } { 4 } , \frac { 3 } { 4 } \right)$
(2) $\left( - \frac { 1 } { 3 } , \frac { 4 } { 3 } \right)$
(3) $\left( - \frac { 1 } { 4 } , \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 3 } { 4 } , \frac { 7 } { 4 } \right)$

Choose the incorrect statement about the two circles whose equations are given below: $x ^ { 2 } + y ^ { 2 } - 10 x - 10 y + 41 = 0$ and $x ^ { 2 } + y ^ { 2 } - 16 x - 10 y + 80 = 0$
(1) Distance between two centres is the average of radii of both the circles.
(2) Both circles' centres lie inside region of one another.
(3) Both circles pass through the centre of each other.
(4) Circles have two intersection points.

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 1 } = 1$ and the circle $x^2 + y^2 = 3$ at their points of intersection. Then $\tan\theta$ is equal to:
(1) $\frac{4}{\sqrt{3}}$
(2) $\frac{2}{\sqrt{3}}$
(3) $2$
(4) $\frac{5}{2\sqrt{3}}$

If the curves, $\frac { x ^ { 2 } } { a } + \frac { y ^ { 2 } } { b } = 1$ and $\frac { x ^ { 2 } } { c } + \frac { y ^ { 2 } } { d } = 1$ intersect each other at an angle of $90 ^ { \circ }$, then which of the following relations is TRUE?
(1) $a - c = b + d$
(2) $a - b = c - d$
(3) $a + b = c + d$
(4) $a b = \frac { c + d } { a + b }$

If the circles $( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = r ^ { 2 }$ and $x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 4 = 0$ intersect at exactly two distinct points, then
(1) $5 < \mathrm { r } < 9$
(2) $0 < \mathrm { r } < 7$
(3) $3 < r < 7$
(4) $\frac { 1 } { 2 } < \mathrm { r } < 7$

If the points of intersection of two distinct conics $x ^ { 2 } + y ^ { 2 } = 4 b$ and $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ lie on the curve $y ^ { 2 } = 3 x ^ { 2 }$, then $3 \sqrt { 3 }$ times the area of the rectangle formed by the intersection points is $\_\_\_\_$ .

Two parabolas have the same focus $(4,3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to:
(1) 392
(2) 384
(3) 192
(4) 96

On the coordinate plane, there is a regular hexagon $A B C D E F$ with side length 3, where $A ( 3,0 ) , D ( - 3,0 )$. How many intersection points does the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 7 } = 1$ have with the regular hexagon $A B C D E F$?
(1) 0
(2) 2
(3) 4
(4) 6
(5) 8

$$|\mathrm{OM}| = 2 \text{ units}$$
In the rectangular coordinate plane, a semicircle with center at point M and a quarter circle with center at the origin intersect at point A as shown in the figure.
Accordingly, what is the x-coordinate of point A?
A) $\frac{5}{3}$ B) $\sqrt{2}$ C) $\frac{\sqrt{3}}{2}$ D) $\frac{3}{2}$ E) $\sqrt{3}$