Position of a Point W.R.T Hyperbola

Q.

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

1. None of these

2. Outside

3. lie on

4. Inside

Q.

Which among the following points lies outside the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$.

1. (4, 3)

2. (9, 3)

3. (4, 3.5)

4. (7, 4)

Q.

If $\frac{x^2}{36}-\frac{y^2}{k^2}=1$ is a hyperbola then which of the following statement can be true?

1. (5, 2) lies on the hyperbola

2. (-3, 1) lies on the hyperbola

3. (3, 1) lies on the hyperbola

4. (10, 4) lies on the hyperbola

Q. Only one real tangent can be drawn from the point (4, 3) to the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$

1. False
2. True

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

1. True
2. False

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

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

Q. The 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

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. If the points $(\sqrt{3}\text{sin}\theta ,\sqrt{4}\text{cos}\theta )$ where $\theta \in R$, lies outside the hyperbola $\frac{x^2}{4}-\frac{y^2}{5}=1$, then the value of $\theta$ is:

1. $\theta \in R$
2. $\theta \in R-\left\{\frac{\pi }{4}\right\}$
3. $\theta \in R-\{\frac{\pi }{2},0\}$
4. $\theta \in R-\{\frac{\pi }{4},0,\frac{\pi }{2}\}$