If the area of the auxiliary circle of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b)$ is twice the area of the ellipse,then the eccentricity of the ellipse is

Two unknown resistances are connected in two gaps of a meter-bridge. The null point is obtained at $40 \ cm$ from left end. A $30 \Omega$ resistance is connected in series with the smaller of the two resistances, the null point shifts by $20 \ cm$ to the right end. The value of smaller resistance in $\Omega$ is

$A$ light beam is travelling from Region $I$ to Region $IV$ (refer figure). The refractive indices in Region $I, II, III$ and $IV$ are $n_0, \frac{n_0}{2}, \frac{n_0}{6}$ and $\frac{n_0}{8}$,respectively. The angle of incidence $\theta$ for which the beam just misses entering Region $IV$ is

The average degree of freedom per molecule of a gas is $6$. The gas performs $25 \ J$ of work while expanding at constant pressure. The heat absorbed by the gas is .... $J$.

Planet $M$ orbits around its sun,$S$,in an elliptical orbit with the sun at one of the foci. When $M$ is closest to $S$,it is $2 \ unit$ away. When $M$ is farthest from $S$,it is $18 \ unit$ away. Assuming $S$ is at the origin $(0, 0)$ and the other focus lies on the negative $y$-axis,find the equation of the elliptical orbit of planet $M$.

Explain why alkyl groups act as electron donors when attached to a $\pi$ system.