Double Ordinate of Hyperbola

Q. The given circle $x^2+y^2+2px=0$, $pϵR$ touches the parabola $y^2=4x$ externally, then

1. p < 0
2. p > 0
3. 0 < p < 1
4. p < - 1

Q. In $\Delta ABC$ prove: $\cos 2A+\cos 2B+\cos 2C=-1-4\cos A\cos B\cos C$

Q. If AB is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $\Delta OAB$ is an equilateral traingle O, being the origin, then the eccentricity of the hyperbola satisfies

1. $1<e\frac{1}{\sqrt{3}}$
2. $e=\frac{2}{\sqrt{3}}$
3. $e>\sqrt{3}$
4. $e>\frac{2}{\sqrt{3}}$

Q.

If AB is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $\Delta OAB$ is an equilateral triangle O being the origin, then the eccentricity of the hyperbola satisfies

1. e >

2. 1 < e <

3. 

4. e >

Q. If $PQ$ is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola, then the eccentricity $e$ of the hyperbola satisfies

1. $1<e<\frac{2}{\sqrt{3}}$
2. $e>\frac{2}{\sqrt{3}}$
3. $e=\frac{2}{\sqrt{3}}$
4. $e>\frac{4}{\sqrt{3}}$

Q. If PQ is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that OPQ is an equilateral triangle, O being the centre of the hyperbola. Then the eccentricity e of the hyperbola, satisfies

1. 
2. 
3. 
4. 

Q. If PQ is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$such that OPQ is an equilateral triangle, O being the centre of the hyperbola. Then, the eccentricity e of the hyperbola satisfies

1. 
   
2. 
3. 
4. 
   

Q. If PQ is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that OPQ is an equilateral triangle, O being the centre of the hyperbola. Then the eccentricity e of the hyperbola, satisfies

1. $\frac{1<e<2}{\sqrt{3}}$
2. $\frac{e=2}{\sqrt{3}}$
3. $\frac{e=\sqrt{3}}{2}$
4. $\frac{e>2}{\sqrt{3}}$

Q. The value of $\lambda$ for which the curve ${(7x+5)}^2+{(7y+3)}^2={\lambda }^2{(4x+3y-24)}^2$ represents a parabola is

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

Q. Find the possible values of $a$ such that $f\left(x\right)=e^{2x}-(a+1)e^x+2x$ is monoatomically increasing for $x\in R$.