For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$, one asymptote is $y = \sqrt { 2 } x$. Then the eccentricity of $C$ is $\_\_\_\_$ .

For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with left and right foci $F _ { 1 } , F _ { 2 }$ respectively, the eccentricity is $\sqrt { 5 }$ . $P$ is a point on $C$ such that $F _ { 1 } P \perp F _ { 2 } P$ . If the area of $\triangle P F _ { 1 } F _ { 2 }$ is 4 , then $a =$
A. $1$
B. $2$
C. $4$
D. $8$

5. Let $F_1, F_2$ be the two foci of hyperbola $C$. Let $P$ be a point on $C$, and $\angle F_1 P F_2 = 60°$, $|PF_1| = 3|PF_2|$. Then the eccentricity of $C$ is
A. $\frac{\sqrt{7}}{2}$
B. $\frac{\sqrt{13}}{2}$
C. $\sqrt{7}$
D. $\sqrt{13}$

3. For the parabola $y ^ { 2 } = 2 p x ( p > 0 )$, the distance from its focus to the line $y = x + 1$ is $\sqrt { 2 }$. Then $p =$
A. 1
B. 2
C. $2 \sqrt { 2 }$
D. 4
【Answer】B 【Solution】 【Analysis】First determine the coordinates of the focus of the parabola, then use the point-to-line distance formula to find the value of $p$.
【Detailed Solution】The focus of the parabola has coordinates $\left( \frac { p } { 2 } , 0 \right)$. The distance from this point to the line $x - y + 1 = 0$ is: $\quad d = \frac { \left| \frac { p } { 2 } - 0 + 1 \right| } { \sqrt { 1 + 1 } } = \sqrt { 2 }$, Solving: $p = 2$ (we discard $p = -6$). Therefore, the answer is: B.

For the ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, let $A$ be the left vertex. Points $P$ and $Q$ are both on $C$ and symmetric about the $y$-axis. If the product of the slopes of $AP$ and $AQ$ is $\frac { 1 } { 4 }$, then the eccentricity of $C$ is:
A. $\frac { \sqrt { 3 } } { 2 }$
B. $\frac { \sqrt { 2 } } { 2 }$
C. $\frac { 1 } { 2 }$
D. $\frac { 1 } { 3 }$

For an ellipse $C$, $F_1, F_2$ are its two foci, $M, N$ are two points on the ellipse. If $\cos \angle F_1NF_2 = \frac{3}{5}$, then the eccentricity of $C$ is
A. $\frac{1}{2}$
B. $\frac{3}{2}$
C. $\frac{\sqrt{13}}{2}$
D. $\frac{\sqrt{17}}{2}$

Given the parabola $y ^ { 2 } = 16 x$, the coordinates of the focus are \_\_\_\_.

Given the hyperbola $\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$, find the slopes of lines passing through $( 3,0 )$ that have only one intersection point with the hyperbola \_\_\_\_.

Let the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ have left and right foci $F _ { 1 } , F _ { 2 }$ respectively. A line through $F _ { 2 }$ parallel to the $y$-axis intersects $C$ at points $A$ and $B$ . If $\left| F _ { 1 } A \right| = 13 , | A B | = 10$ , then the eccentricity of $C$ is $\_\_\_\_$ .

If the imaginary axis length of hyperbola $C$ is $\sqrt{7}$ times the real axis length, then the eccentricity of $C$ is
A. $\sqrt{2}$
B. $2$
C. $\sqrt{7}$
D. $2\sqrt{2}$

For the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, let $F_1, F_2$ be the left and right foci respectively, and $A_1, A_2$ be the left and right vertices respectively. The circle with diameter $F_1F_2$ intersects one asymptote of $C$ at points $M, N$, and $\angle NA_1M = \frac{5\pi}{6}$, then
A. $\angle A_1MA_2 = \frac{\pi}{6}$
B. $|MA_1| = 2|MA_2|$
C. The eccentricity of $C$ is $\sqrt{13}$
D. When $a = \sqrt{2}$, the area of quadrilateral $NA_1MA_2$ is $8\sqrt{3}$

If the imaginary axis length of hyperbola $C$ is $\sqrt{7}$ times the real axis length, then the eccentricity of $C$ is
A. $\sqrt{2}$
B. $2$
C. $\sqrt{7}$
D. $2\sqrt{2}$

The line $2 \mathrm { x } + \mathrm { y } = 1$ is tangent to the hyperbola $\frac { \mathrm { x } ^ { 2 } } { \mathrm { a } ^ { 2 } } - \frac { \mathrm { y } ^ { 2 } } { \mathrm {~b} ^ { 2 } } = 1$. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

Let $P ( 6,3 )$ be a point on the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If the normal at the point $P$ intersects the $x$-axis at $( 9,0 )$, then the eccentricity of the hyperbola is
(A) $\sqrt { \frac { 5 } { 2 } }$
(B) $\sqrt { \frac { 3 } { 2 } }$
(C) $\sqrt { 2 }$
(D) $\sqrt { 3 }$

If $2x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{16} = 1$, then which of the following CANNOT be sides of a right angled triangle?
[A] $a, 4, 1$
[B] $a, 4, 2$
[C] $2a, 8, 1$
[D] $2a, 4, 1$

Let $H$ : $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, where $a > b > 0$, be a hyperbola in the $x y$-plane whose conjugate axis $L M$ subtends an angle of $60 ^ { \circ }$ at one of its vertices $N$. Let the area of the triangle $L M N$ be $4 \sqrt { 3 }$.
LIST-I P. The length of the conjugate axis of $H$ is Q. The eccentricity of $H$ is R. The distance between the foci of $H$ is S. The length of the latus rectum of $H$ is
LIST-II

1. 8
2. $\frac { 4 } { \sqrt { 3 } }$
3. $\frac { 2 } { \sqrt { 3 } }$
4. 4

The correct option is:
(A) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(B) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 3 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 2 }$
(C) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 1 } ; \mathbf { R } \rightarrow \mathbf { 3 } ; \mathbf { S } \rightarrow \mathbf { 2 }$
(D) $\mathbf { P } \rightarrow \mathbf { 3 } ; \mathbf { Q } \rightarrow \mathbf { 4 } ; \mathbf { R } \rightarrow \mathbf { 2 } ; \mathbf { S } \rightarrow \mathbf { 1 }$

Let $a , b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y ^ { 2 } = 4 \lambda x$, and suppose the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is
(A) $\frac { 1 } { \sqrt { 2 } }$
(B) $\frac { 1 } { 2 }$
(C) $\frac { 1 } { 3 }$
(D) $\frac { 2 } { 5 }$

If $OB$ is the semi-minor axis of an ellipse, $F _ { 1 }$ and $F _ { 2 }$ are its focii and the angle between $F _ { 1 } B$ and $F _ { 2 } B$ is a right angle, then the square of the eccentricity of the ellipse is
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 2 \sqrt { 2 } }$

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is: (1) $\frac{4}{3}$ (2) $\frac{4}{\sqrt{3}}$ (3) $\frac{2}{\sqrt{3}}$ (4) $\sqrt{3}$

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is $\frac { 3 } { 2 }$ units, then its eccentricity is
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 9 }$
(4) $\frac { 1 } { 3 }$

A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length 4 along the $x$-axis. Then the eccentricity of the hyperbola is:
(1) $\sqrt{3}$
(2) $\frac{3}{2}$
(3) $\frac{2}{\sqrt{3}}$
(4) 2

If the distance between the foci of an ellipse is 6 and the distance between its directrix is 12, then the length of its latus rectum is
(1) $\sqrt { 3 }$
(2) $3 \sqrt { 2 }$
(3) $\frac { 3 } { \sqrt { 2 } }$
(4) $2 \sqrt { 3 }$

The length of the minor axis (along $y$-axis) of an ellipse in the standard form is $\frac { 4 } { \sqrt { 3 } }$. If this ellipse touches the line $x + 6 y = 8$ then its eccentricity is:
(1) $\frac { 1 } { 2 } \sqrt { \frac { 11 } { 3 } }$
(2) $\sqrt { \frac { 5 } { 6 } }$
(3) $\frac { 1 } { 2 } \sqrt { \frac { 5 } { 3 } }$
(4) $\frac { 1 } { 3 } \sqrt { \frac { 11 } { 3 } }$

If $e _ { 1 }$ and $e _ { 2 }$ are the eccentricities of the ellipse $\frac { x ^ { 2 } } { 18 } + \frac { y ^ { 2 } } { 4 } = 1$ and the hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ respectively and $\left( e _ { 1 } , e _ { 2 } \right)$ is a point on the ellipse $15 x ^ { 2 } + 3 y ^ { 2 } = k$, then the value of $k$ is equal to
(1) 16
(2) 17
(3) 15
(4) 14

For some $\theta \in \left( 0 , \frac { \pi } { 2 } \right)$, if the eccentricity of the hyperbola, $x ^ { 2 } - y ^ { 2 } \sec ^ { 2 } \theta = 10$ is $\sqrt { 5 }$ times the eccentricity of the ellipse, $x ^ { 2 } \sec ^ { 2 } \theta + y ^ { 2 } = 5$, then the length of the latus rectum of the ellipse, is
(1) $2 \sqrt { 6 }$
(2) $\sqrt { 30 }$
(3) $\frac { 2 \sqrt { 5 } } { 3 }$
(4) $\frac { 4 \sqrt { 5 } } { 3 }$