Standard Equation of Hyperbola

Q. The equation $\frac{x^2}{9-\lambda }+\frac{y^2}{4-\lambda }=1$ represents a hyperbola when $a<\lambda <b$, then value of $\left[\frac{b+a}{b-a}\right]$(Where [.] denotes greatest integer function) is

Q. The equation represents $x^2-2xy+y^2+3x+2=0$ represents

1. A parabola
2. An ellipse
3. A hyperbola
4. A circle

Q. If a hyperbola has length of its conjugate axis equal to $5$ and the distance between its foci is $13$, then the eccentricity of the hyperbola is :

1. $\frac{13}{6}$
2. $2$
3. $\frac{13}{8}$
4. $\frac{13}{12}$

Q.

Find the equation of the tangent plane at $(1,1,1)$ to $x^2+y^2+z^2-3=0$.

Q. Let $S=\left\{\right(x,y)\in R^2:\frac{y^2}{1+r}-\frac{x^2}{1-r}=1\}$ where $r\ne \pm 1$. Then $S$ represents :

1. A ellipse whose eccentricity is $\sqrt{\frac{2}{r+1}}$, when $r>1$
2. A ellipse whose eccentricity is $\frac{1}{\sqrt{r+1}}$, when $r>1$
3. A hyperbola whose eccentricity is $\sqrt{\frac{2}{1-r}}$, when $-1<r<1$
4. A hyperbola whose eccentricity is $\sqrt{\frac{2}{r+1}}$, when $-1<r<1$

Q.

The two points A and B in a plane are such that for all points P lies on circle satisfied $\frac{PA}{PB}=k$, then k will not be equal to

1. 0

2. 2

3. 1

4. 3

Q. Let $C$ be the curve(s) given by $(a{x}^2+b{y}^2+c)(a{y}^2-bx)=0;a,b,c\in R-\left\{0\right\}$ and $a\ne b.$ Then $C$ represents

1. a parabola only if $a,b$ and $c$ are of the same sign.
2. a circle or a parabola if $a$ and $b$ are of the same sign and $c$ is of opposite sign.
3. an ellipse or a parabola if $a$ and $b$ are of the same sign and $c$ is of opposite sign.
4. a hyperbola or a parabola if $a$ and $b$ are of opposite sign

Q.

The equation of the hyperbola referred to its axes as coordinate axes, whose latus rectum is 4 and eccentricity is 3 is

1. 

2. 

3. 

4. 

Q. Equation of tangent to the hyperbola $2x^2-3y^2=6$ which is parallel to the line y = 3x + 4, is

1. y = 3x + 5
2. y = 3x - 5
3. y = 3x + 5 and y = 3x - 5
4. None of these

Q.

If P is any point on the hyperbola whose axis are equal, prove that SP.SP=$C{P}^2$

Q.

$\text{Find the equation of the hyperbola whose}\left(i\right)\text{foci are (6, 4)and (-4, 4) and eccentricity is 2.}\left(ii\right)\text{vertices are (-8, -1) and (16, -1) and focus is(17, -1)}\left(iii\right)\text{foci are (4, 2) and (8, 2) and eccentricity is 2.}\left(iv\right)\text{vertices are at}(0\pm 7)andfociat(0,\pm \frac{28}{4}).$

Q.

How do you find the major axis of an ellipse?

Q.

Equation of the hyperbola with length of the latusrectum $\frac{9}{2}$ and e = $\frac{5}{4}$ is

1. 

2. 

3. 

4. 

Q. The solution of the differential equation $2x\frac{dy}{dx}-y=0;y\left(1\right)=2$ represents _______

1. ellipse
2. parabola
3. circle
4. straight line

Q. If $e$ and $e^{\prime }$ be the eccentricities of a hyperbola and its conjugate, them $\frac{1}{e^2}+\frac{1}{e^{\prime 2}}$ is equal to :

1. 0
2. 1
3. None of these
4. 2

Q. If $\alpha ={\int }_0^1\left(e^{(9x+3{\tan }^{-1}x)}\right)\left(\frac{12+9x^2}{1+x^2}\right)dx$ , where ${\tan }^{-1}x$ takes only principal values, then the value of $({\log }_e|1+\alpha |-\frac{3\pi }{4})$ is

Q.

Find the equation of the sphere with center $(3,6,-4)$ and touching the plane $2x-2y-z-10=0$?

Q. If $x\cos \alpha +y\sin \alpha =p$, where $p=(-\frac{{\sin }^2\alpha }{\cos \alpha })$ be a straight line, then perpendiculars on this straight line from the points $(m^2,2m),(m^{\prime }m,m+m^{\prime }),(m^{\prime 2}m,2m^{\prime })$ form a G.P.

1. True
2. False

Q. The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through the point $(\sqrt{6},3)$ and the length of the latusrectum is $\frac{18}{5}$. Then, the length of the transverse axis is equal to

1. $3$
2. $5$
3. $4$
4. $2$
5. $1$

Q.
Consider the functions defined implicitly by the equation $y^3-3y+x=0$ on various intervals in the real line. If $x\in (-\infty ,-2)\cup (2,\infty )$, the equation implicitly defines a unique real valued differentiable function $y=f\left(x\right)$. If $x\in (-2,2)$, the equation implicitly defines a unique real valued differentiable function $y=g\left(x\right)$ satisfying $g\left(0\right)=0$.

1. $-{\int }_a^b\frac{x}{3\left(\right(f\left(x\right){)}^2-1)}dx+bf\left(b\right)-af\left(a\right)$
2. ${\int }_a^b\frac{x}{3\left(\right(f\left(x\right){)}^2-1)}dx-bf\left(b\right)+af\left(a\right)$
3. $-{\int }_a^b\frac{x}{3\left(\right(f\left(x\right){)}^2-1)}dx-bf\left(b\right)+af\left(a\right)$
4. ${\int }_a^b\frac{x}{3\left(\right(f\left(x\right){)}^2-1)}dx+bf\left(b\right)-af\left(a\right)$

Q.

Equation of the hyperbola with foci (0, ± 5) and e = $\frac{5}{3}$ is

1. $\frac{y^z}{9}-\frac{y^z}{16}=1$

2. $\frac{x^z}{16}-\frac{y^z}{9}=1$

3. $\frac{x^z}{16}-\frac{y^z}{9}=1$

4. $\frac{x^z}{12}-\frac{y^z}{13}=1$

Q. The equation of the hyperbola whose foci are $(6,5),(-4,5)$ and eccentricity $5/4$ is?

1. $\frac{(x-1{)}^2}{16}-\frac{(y-5{)}^2}{9}=1$
2. $\frac{x^2}{16}-\frac{y^2}{9}=1$
3. $\frac{(x-1{)}^2}{16}-\frac{(y-5{)}^2}{9}=-1$
4. $\frac{(x-1{)}^2}{4}-\frac{(y-5{)}^2}{9}=1$

Q.

For hyperbola $\frac{x^2}{144}-\frac{y^2}{100}=1$, circle $x^2+y^2=169$, concyclic points are

1. $\pm \frac{\sqrt{269}}{\sqrt{61}},\pm \frac{5}{\sqrt{61}}$
2. $\pm \frac{6\sqrt{29}}{\sqrt{61}},\pm \frac{25}{\sqrt{61}}$
3. $\pm \frac{6\sqrt{269}}{\sqrt{61}},\pm \frac{25}{\sqrt{61}}$
4. None

Q. Circles are drawn on chords of the rectangular hyperbola $xy=a^2$ 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. Circles are drawn on chords of the rectangular hyperbola $xy=4$ parallel to the line $y=x$ as diameters.All such circles pass through two fixed points whose coordinates are

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

Q. Calculate the area bounded by the curve $y(y-1)$ and the y-axis ?

Q. The Vertex of the parabola $y^2-10y+x+22=0$ is.

1. (3, 4)
2. (3, 5)
3. (5, 3)
4. none of these

Q.

For the hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$

Which one of the following remain constant with change of $\alpha$ ?

1. Abscissa of vertices

2. abscissa of foci

3. Eccentricity

4. directrix

Q. Find the equation of the hyperbola whose foci are $(0,\pm \sqrt{10})$ and passing through the point $(2,3)$.

Q. The differential equation $(3x+4y+1)dx+(4x+5y+1)dy=0$ represents a family of

1. Parabolas
2. Hyperbolas
3. Ellipses
4. Circles