Standard Equation of Hyperbola

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.

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. 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. Choose the correct statement(s) regarding standard hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$

1. Line perpendicular to the line joining A and A' and passing through centre is called conjugate axis.
2. Line joining vertices A and A' is called conjugate axis.
3. Line perpendicular to the line joining A and A' is called transverse axis.
4. Line joining vertices A and A' is called transverse axis.

Q.

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

1. $\frac{x^z}{45}-\frac{y^z}{36}=1$

2. $\frac{x^z}{16}-\frac{y^z}{45}=1$

3. $\frac{x^z}{16}-\frac{y^z}{32}=1$

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

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.

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

1. $\frac{x^z}{45}-\frac{y^z}{36}=1$

2. $\frac{x^z}{16}-\frac{y^z}{45}=1$

3. $\frac{x^z}{16}-\frac{y^z}{32}=1$

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

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 $-1<r<1$