General Equation of Hyperbola

Q. 30. the equation of the hyperbola whose asymptotes are the straight lines 3x-4y+7=0 and 4x+3y+1=0 which passes through (0, 0)

Q. If the length of latus rectum of a hyperbola $\frac{x^2}{k}-\frac{y^2}{25}=-1$ is $\frac{22}{5}$ units, then its $e$ (eccentricity) is

1. $\frac{7}{3}$
2. $\frac{7}{5}$
3. $\frac{6}{5}$
4. $\frac{7}{2}$

Q.

Find the equation for the ellipse that satisfies the given conditions,
$b=3,c=4,$ centre at origin; foci on the x - axis.

Q. $A(-2,0)$ and $B(2,0)$ are the two fixed points and $P$ is a point such that $PA-PB=2$. Let $S$ be the circle $x^2+y^2=r^2$, then match the following.

$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. If}r=2,\text{then the number of points}P\text{satisfying}& \text{p. 2}\\ PA-PB=2\text{and lying on}{x}^2+y^2=r^2\text{is}& \\ \text{b. If}r=1,\text{then the number of points}P\text{satisfying}& \text{q. 4}\\ PA-PB=2\text{and lying on}{x}^2+y^2=r^2\text{is}& \\ \text{c. For}r=2\text{the number of common tangents is}& \text{r.}0\\ \text{d. For}r=\frac{1}{2}\text{the number of common tangents is}& \text{s.}1\end{array}$

1. $a-q;b-p;c-r;d-p$
2. $a-s;b-q;c-r;d-p$
3. $a-p;b-s;c-r;d-p$
4. $a-q;b-p;c-s;d-s$

Q. If a variable line has its intercepts on the co-ordinate axes $e,e^{\prime }$, where $\frac{e}{2},\frac{e^{\prime }}{2}$ are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle $x^2+y^2=r^2$, where $r=$

Q.

Equation of the hyperbola with focus (-3, 4) directrix 3x-4y+5=0 and e = $\frac{5}{2}$ is

1. $5x^2-24xy+12y^2+6x-8y-75=0$

2. $5x^2-24xy+12y^2-8x-6y-25=0$

3. $5x^2-24xy+12y^2-12x+8y-55=0$

4. $5x^2-24xy+12y^2-7x-12y-65=0$

Q. If the chord $x\cos \alpha +y\sin \alpha =p$ of the hyperbola $\frac{x^2}{16}-\frac{y^2}{18}=1$ subtends a right angle at the centre, and the diameter of the circle, concentric with the hyperbola to which the given chord is a tangent is $d$ units, then the value of $\frac{d}{4}$ is

Q. If the length of latus rectum of a hyperbola whose eccentricity is $\frac{3}{\sqrt{5}}$, centre $(4,3)$ and axis is parallel to coordinate axis is $8$ units, then the equation(s) of the hyperbola is/are

1. $\frac{(y-3{)}^2}{25}-\frac{(x-4{)}^2}{20}=1$
2. $\frac{(x-4{)}^2}{25}-\frac{(y-3{)}^2}{20}=1$
3. $\frac{(x-4{)}^2}{25}-\frac{(y-3{)}^2}{15}=1$
4. $\frac{(2x+y-11{)}^2}{25}-\frac{(x-2y+2{)}^2}{15}=1$