Position of a Point W.R.T Hyperbola

Q. If the line $5x+12y=9$ touches the hyperbola $x^2-9y^2=9$, then its point of contact is

1. $(5,-\frac{4}{3})$
2. $(5,\frac{4}{3})$
   
3. $(-3,2)$
4. $(3,0)$

Q. A hyperbola passes through the point $P(\sqrt{2},\sqrt{3})$ and has focii at $(\pm 2,0).$ Then the tangent to this hyperbola at $P$ also passes through the point:

1. $(3\sqrt{2},2\sqrt{3})$
2. $(2\sqrt{2},3\sqrt{3})$
3. $(\sqrt{3},\sqrt{2})$
4. $(-\sqrt{2},-\sqrt{3})$

Q. The number of tangents to the hyperbola $\frac{x^2}{4}-\frac{y^2}{3}=1$ through $(1,4)$ is

Q. The coordinates of a point on the hyperbola, $\frac{x^2}{24}-\frac{y^2}{18}=1,$ which is nearest to the line $3x+2y+1=0$ are

1. $(6,3)$
2. $(-6,-3)$
3. $(6,-3)$
4. $(-6,3)$

Q. If the line $5x+12y=9$ touches the hyperbola $x^2-9y^2=9$, then its point of contact is

1. $(5,-\frac{4}{3})$
2. $(5,\frac{4}{3})$
   
3. $(-3,2)$
4. $(3,0)$

Q. Which among the following point lie inside the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$

1. $(3,1)$
2. $(5,3)$
3. $(4,5)$
4. $(1,0)$

Q. The equation of common tangent(s) to the hyperbola $9x^2-16y^2=144$ and circle $x^2+y^2=9$ is/are

1. $\sqrt{7}y=-2\sqrt{2}x-15$
2. $\sqrt{7}y=3\sqrt{2}x+15$
3. $\sqrt{7}y=2\sqrt{2}x+15$
4. $\sqrt{7}y=-3\sqrt{2}x-15$

Q. The point (5, -4) lies inside the hyperbola $9x^2-y^2=1$.

1. True
2. False

Q. The position of the point $(5,-4)$ relative to the hyperbola $9y^2-x^2=1$ is :

1. inside
2. outside
3. on the hyberbola
4. on the directrix of hyperbola