Focii of Hyperbola

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.

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

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