Focii of Hyperbola

Q. The area of the trapezium whose vertices lie on the parabola $y^2=4x$ and diagonals pass through $(1,0)$ is $k$ units. If the length of the diagonals is $\frac{25}{4}$ units each, then the value of $4k$ is

Q.

Consider a branch of the hyperbola $x^2-2y^2-2\sqrt{2}x-4\sqrt{2}y-6=0$ with vertex at the point A. Let B be one of the end points of its latusrectum. If C is the focus of the hyperbola nearest to the point A, then the area of the $\Delta ABC$ is

1. $1-\sqrt{\frac{2}{3}}$ sq unit

2. $\sqrt{\frac{3}{2}}-1$ sq unit

3. $1+\sqrt{\frac{2}{3}}$ sq unit

4. $\sqrt{\frac{3}{2}}+1$ sq unit

Q. For the hyperbola $\frac{(3x-4y-12{)}^2}{100}-\frac{(4x+3y-12{)}^2}{225}=1$

1. coordinates of center is$(\frac{84}{25},\frac{-12}{25})$
2. coordinates of center is $(\frac{12}{25},\frac{84}{25})$
3. coordinates of foci is $(\frac{84\pm 100\sqrt{13}}{25},\frac{-12\mp 75\sqrt{13}}{25})$
4. coordinates of foci is $(\frac{84\pm 15\sqrt{13}}{25},\frac{-12\mp 20\sqrt{13}}{25})$

Q.

For hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$, which of the following remains constant with change in ${}^{\prime }{\alpha }^{\prime }$ ?

1. Abscissae of vertices

2. Abscissae of foci

3. Eccentricity

4. Directrix

Q.

If P is a point on the hyperbola $16x^2-9y^2=144$ whose foci are $S_{1}and{S}_2,thenP{S}_1\sim P{S}_2=$

1. 6

2. 8

3. 4

4. 12

Q.

$Thefociofthehyperbola2x^2-3y^2=5are$

1. $(\pm \sqrt{5}/6,0)$

2. $(\pm 5/\sqrt{6}.0)$

3. $(\pm 5/6.0)$

4. none of these

Q. An ellipse passes through the foci of the hyperbola, $9x^2–4y^2=36$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $\frac{1}{2},$ then which of the following points $doesnot$ lie on the ellipse?

1. $(\sqrt{\frac{13}{2}},\sqrt{6})$
2. $(\frac{\sqrt{39}}{2},\sqrt{3})$
3. $(\sqrt{13},0)$
4. $(\frac{1}{2}\sqrt{13},\frac{\sqrt{3}}{2})$

Q. If pair of straight lines $x^2-y^2+6x+4y+5=0$ are transverse and conjugate axes of hyperbola and perpendicular distance form origin to these lines represent the length of transverse and conjugate axis, then the eccentricity of hyperbola is:

1. $\sqrt{3}$
2. $\frac{\sqrt{26}}{5}$
3. $\sqrt{26}$
4. $2\sqrt{13}$

Q.

Which of the following is INCORRECT for the hyperbola $x^2-2y^2-2x+8y-1=0$

1. Its eccentricity is

2. Length of the transverse axis is

3. Length of the conjugate axis is

4. Latus rectum

Q. A hyperbola, having the transverse axis of length $2\sin \theta$ is confocal with the ellipse $3x^2+4y^2=12.$ Its equation is

1. $x^2{\text{cosec}}^2\theta -y^2{\sec }^2\theta =1$
2. $x^2{\sin }^2\theta -y^2{\cos }^2\theta =1$
3. $(x^2+y^2){\sin }^2\theta =1+y^2$
4. $x^2{\text{cosec}}^2\theta =x^2+y^2+{\sin }^2\theta$

Q.

If P is a point on the hyperbola $16x^2-9y^2=144$ whose foci are $S_{1}and{S}_2,thenP{S}_1\sim P{S}_2=$

1. 4

2. 6

3. 8

4. 12

Q.

The foci of the hyperbola$9x^2-16y^2=144$ are

1. $(\pm 4,0)$

2. $(0,\pm 4)$

3. $(\pm 5,0)$

4. $(0,\pm 5)$

Q. The normal at any point $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ meets the x-axis at $A$ and y-axis at $B$. If $PA:PB=1:4,$ then the eccentricity of the ellipse is

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

Q. For the hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$, which of the following remains constant when α varies?

1. Eccentricity
2. Abscissae of vertices
3. Directrix
4. Abscissae of foci

Q. Circles are drawn on chords of the rectangular hyperbola $xy=c^2$ parallel to the line $y=x$ as diameters. All such circles pass through two fixed points whose co-ordinates are

1. $(-c,-c)$
2. $(c,c)$
3. $(-c,c)$
4. $(c,-c)$

Q.

$\text{Write the coordinates of the foci of the hyperbola}9x^2-16y^2=144$

Q. If a hyperbola has one focus at the origin and its eccentricity is $\sqrt{2}$ . One of the directrices is $x+y+1=0.$Then the centre of the hyperbola is

1. 
2. 
3. 
4. 

Q.

Consider a branch of the hyperbola $x^2-2y^2-2\sqrt{2}x-4\sqrt{2}y-6=0$ with vertex at the point A. Let B be one of the end points of its latusrectum. If C is the focus of the hyperbola nearest to the point A, then the area of the $\Delta ABC$ is

1. $1-\sqrt{\frac{2}{3}}$ sq unit

2. $\sqrt{\frac{3}{2}}-1$ sq unit

3. $1+\sqrt{\frac{2}{3}}$ sq unit

4. $\sqrt{\frac{3}{2}}+1$ sq unit

Q. If the parabola $y=-x^2-2x+k$ touches the parabola $y=-\frac{1}{2}{x}^2-4x+3$, then the value of $k$ is

1. $2$
2. $1$
3. $3$
4. $4$

Q. Assertion(A): The difference of the focal distances of any point on the hyperbola $\frac{x^2}{36}-\frac{y^2}{9}=1$ is 12.
Reason(R): The difference of the focal distances of any point on the hyperbola is equal to the length of it transverse axis

1. Both A and R are true and R is the correct
   explanation of A.
2. Both A and R are true but R is not the correct
   explanation of A.
3. A is true but R is false.
4. A is false but R is true.

Q. Let $nϵN$ If ${(1+x)}^n=a_0+a_{1}x+a_2{x}^2+\cdot \cdot +a_n{x}^n$ and $a_{n-3},a_{n-2},a_{n-1}$ are in AP then

1. $a_1,a_2,a_3$ are in AP
2. $a_1,a_2,a_3$ are in HP
3. n=7
4. n=14

Q. For three non coplanar vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ the relation $|\overrightarrow{a}\times \overrightarrow{b}\cdot \overrightarrow{c}|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|$ holds if and only if

1. $\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0$
2. $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0$
3. $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=0$
4. $\overrightarrow{c}.\overrightarrow{a}=\overrightarrow{a}.\overrightarrow{b}=0$

Q. The coordinates of the foci of the hyperbola $xy=c^2$ are

1. $(\pm C,\pm C)$
2. $(\pm \frac{c}{\sqrt{2}}<\pm \frac{c}{\sqrt{2}})$
3. $(\pm 2c,\pm 2c)$
4. $(\pm \sqrt{2}c,\pm \sqrt{2}c)$

Q. Find the length of the side of an equilateral triangle inscribed in the parabola, $y^2=4x$ so that one of its angular point is at the vertex.

1. $16\sqrt{3}$
2. $4\sqrt{3}$
3. $8\sqrt{3}$
4. $2\sqrt{3}$

Q.

The distance between the foci of the hyperbola $9x^2-16y^2+18x+32y-151$ = 0 is

1. 8

2. 10

3. 6

4. 2

Q.

For hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$, which of the following remains constant with change in ${}^{\prime }{\alpha }^{\prime }$ ?

1. Eccentricity

2. Directrix

3. Abscissae of vertices

4. Abscissae of foci

Q.

For hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$, which of the following remains constant with change in ${}^{\prime }{\alpha }^{\prime }$ ?

1. Abscissae of vertices

2. Abscissae of foci

3. Eccentricity

4. Directrix

Q. Origin is the center of circle passing through the vertical of an equilateral triangle whose median is of length 3a then equation of the circle is

1. $x^2+y^2={3a}^2$
2. $x^2+y^2={2a}^2$
3. $x^2+y^2=a^2$
4. $x^2+y^2={4a}^2$

Q. An ellipse passes through the foci of the hyperbola, $9x^2–4y^2=36$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $\frac{1}{2},$ then which of the following points $doesnot$ lie on the ellipse?

1. $(\frac{1}{2}\sqrt{13},\frac{\sqrt{3}}{2})$
2. $(\sqrt{13},0)$
3. $(\sqrt{\frac{13}{2}},\sqrt{6})$
4. $(\frac{\sqrt{39}}{2},\sqrt{3})$

Q. If pair of straight lines $x^2-y^2+6x+4y+5=0$ are transverse and conjugate axes of hyperbola and perpendicular distance form origin to these lines represent the length of transverse and conjugate axis, then the eccentricity of hyperbola is:

1. $\sqrt{3}$
2. $\frac{\sqrt{26}}{5}$
3. $\sqrt{26}$
4. $2\sqrt{13}$