Position of a Line W.R.T Hyperbola

Q. The value of m for which y = mx + 6 is a tangent to the hyperbola $\frac{x^2}{100}-\frac{y^2}{49}=1$ is .

1. $\frac{\sqrt{17}}{10}$
2. $\frac{\sqrt{17}}{20}$
3. $\sqrt{\frac{17}{10}}$
4. $\sqrt{\frac{17}{20}}$

Q.

The line $y=mx+c$ becomes a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then the value of c is

1. $\pm \sqrt{a^2{m}^2+b^2}$

2. $\pm \sqrt{a^2{m}^2-b^2}$

3. $\pm \sqrt{a^2{m}^2+a^2}$

4. $\pm \sqrt{a^2{m}^2-a^2}$

Q.

Length of the straight line $x-3y=1$ intercepted by the hyperbola $x^2-4y^2=1$ is

1. $\sqrt{10}$

2. $\frac{6}{5}$

3. $\frac{1}{\sqrt{10}}$

4. $\frac{6}{5}\sqrt{10}$

Q. A perpendicular is dropped on the transverse axis of the hyperbola $\frac{x^2}{49}-\frac{y^2}{36}=1$ from its asymptote. What will be the product of the segments of this line intercepted between the point and the curve?

1. $49$
2. $7$
3. $36$
4. $6$

Q. The sides $AC$ and $AB$ of a $△ABC$ touches the conjugate hyperbola of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at $C$ and $B.$ If the vertex $A$ lies on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then the side $BC$ always touches

1. $\text{a parabola}$
2. $\text{a circle}$
3. $\text{a hyperbola}$
4. $\text{an ellipse}$

Q.

If the line y=mx+1 touches the hyperbola $\frac{x^z}{9}-\frac{y^z}{2}$ = 1 then m =

1. ± $\frac{1}{\sqrt{2}}$

2. ± $\frac{1}{3}$

3. ± $\frac{1}{\sqrt{3}}$

4. ± $\frac{1}{2}$

Q.

The condition that the line $\frac{x}{p}+\frac{y}{p}=1$ o be tangent to $\frac{x^z}{a^z}+\frac{y^z}{b^z}=1$ is

1. $\frac{a^z}{p^z}-\frac{b^z}{q^z}$ = 1

2. $\frac{a^z}{p^z}-\frac{b^z}{q^z}$ = 1

3. $\frac{a^z}{q^z}-\frac{b^z}{p^z}$ = 1

4. $\frac{a^z}{q^z}-\frac{b^z}{p^z}$ = 1

Q. $\mathrm{The}\mathrm{value}\left(s\right)\mathrm{of}\lambda \mathrm{for}\mathrm{which}\mathrm{the}\mathrm{line\; y=2x+\lambda }$touches the hyperbola $16x^2-9y^2=144$is/are:

1. $\sqrt{5}$
2. -$\sqrt{5}$
3. $2\sqrt{5}$
4. $-2\sqrt{5}$

Q. If $A\left(\theta \right)$ and $B\left(\varphi \right)$ are the parametric ends of a focal chord of $\frac{x^2}{144}-\frac{y^2}{25}=1$, then the maximum value of $\left|\tan \frac{\theta }{2}\tan \frac{\varphi }{2}\right|$ is

Q. If the maximum value of $(x+y{)}^2\text{is}\lambda$ and $P(x,y)$ satisfies $x^2+y^2=1$, then the number of tangents that can drawn from $(\lambda ,0)$ to the hyperbola $(x-2{)}^2-y^2=1$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. If $A\left(\theta \right)$ and $B\left(\varphi \right)$ are the parametric ends of a focal chord of $\frac{x^2}{144}-\frac{y^2}{25}=1$, then the maximum value of $\left|\tan \frac{\theta }{2}\tan \frac{\varphi }{2}\right|$ is