Focal Chord of Hyperbola

Q. If a chord of the parabola $y^2=4x$ passes through its focus and makes an angle $\theta$ the $X-$axis, then its length is

1. $4{\sin }^2\theta$
2. $4{\cos }^2\theta$
3. $4{\text{cosec}}^2\theta$
4. $4{\sec }^2\theta$

Q. The positive value of the parameter 'k' for which the area of the figure bounded by the curve $y=\sin \left(kx\right),x=\frac{2\pi }{3k},x=\frac{5\pi }{3k}$ and x-axis is less than 2 can be

1. $\frac{1}{8}<k<\frac{3}{8}$
2. $0<k<\frac{1}{8}$
3. 1 < k < 2
4. $\frac{3}{8}<k<\frac{5}{8}$

Q. If $\alpha$ and $\beta$ are the eccentric angles of the extremities of a focal chord of an ellipse of eccentricity e then $\cos \left(\frac{\alpha -\beta }{2}\right)=$

1. $e\cos \left(\frac{\alpha -\beta }{2}\right)$
2. $e\cos \left(\frac{\alpha +\beta }{2}\right)$
3. $e\cos \left(\frac{\alpha -\beta }{3}\right)$
4. $e\cos \left(\frac{2\alpha +2\beta }{2}\right)$

Q. If $\alpha$ and $\beta$ are the eccentric angles of the extremities of a focal chord of an ellipse of eccentricity e then $\cos \left(\frac{\alpha -\beta }{2}\right)=$

1. 
2. 
3. 
4.