Parametric Representation: Ellipse

Q. If $\frac{x}{a}+\frac{y}{b}=\sqrt{2}$ touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then its eccentric angle $\theta$ is equal to

1. 
2. 
3. 
4. 

Q. $\alpha ,\beta ,\gamma$ are the parametric angles of three points A, B, C respectively on the circle $x^2+y^2=1$ and P(-1, 0). If the lengths of the chords PA, PB, PC are in G.P. then $cos\left(\frac{\alpha }{2}\right),cos\left(\frac{\beta }{2}\right),cos\left(\frac{\gamma }{2}\right)$ are in

1. A.P
2. G.P
3. H.P
4. A.G.P

Q. The area of the parallelogram formed by the tangents at the points whose eccentric angles are $\theta ,\theta +\frac{\pi }{2},\theta +\pi ,\theta +\frac{3\pi }{2}$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

1. ab
2. 4ab
3. 3ab
4. 2ab

Q. $P\left(\theta \right)$ and $Q(\theta +\frac{\pi }{2})$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$ The locus of midpoint of the chord PQ is

1. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{a}$
2. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{b}$
3. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{2}$
4. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{6}$

Q. The equation with the parametric equations x = 1 + cos $\theta$, y = 2 + 3 sin $\theta$ represents an ellipse.

1. False
2. True

Q. The area of the parallelogram formed by the tangents at the points whose eccentric angles are $\theta ,\theta +\frac{\pi }{2},\theta +\pi ,\theta +\frac{3\pi }{2}$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

1. ab
2. 4ab
3. 3ab
4. 2ab

Q.

If $\cos \alpha =\frac{2}{3},$ then the range of values of $\varphi$ on the ellipse
$x^2+4y^2=4$ falls inside the circle $x^2+y^2+4x+3=0$ is

1. $-\alpha ,\alpha$

2. $0,\alpha$

3. $\alpha ,\pi$

4. $\pi -\alpha ,\pi +\alpha$

Q. The maximum length of chord of the ellipse $\frac{x^2}{8}+\frac{y^2}{4}=1$ such that eccentric angles of its extremities differ by $\frac{\pi }{2},$ is