If the pair of straight lines $6x^2 - 5xy + y^2 = 0$ makes angles $\alpha$ and $\beta$ with the $X$-axis,then $\tan(\alpha - \beta) = $

If the temperature of the sun were to increase from $T$ to $2T$ and its radius from $R$ to $2R$,then the ratio of the radiant energy received on earth to what it was previously will be

The equation of the hyperbola which passes through the point $(2,3)$ and has the asymptotes $4x+3y-7=0$ and $x-2y-1=0$ is

In an astronomical telescope in normal adjustment,a straight black line of length $L$ is drawn on the inside part of the objective lens. The eyepiece forms a real image of this line. The length of this image is $I$. The magnification of the telescope is

In free space,a shell moving with velocity $60\,m/s$ along the positive $x$-axis explodes into two pieces of mass ratio $1:2$ when it passes through the origin. After the explosion,the velocity of the centre of mass is ........ $m/s$.

$A$ line segment $AM = a$ moves in the $XOY$ plane such that $AM$ is parallel to the $X$-axis. If $A$ moves along the circle $x^2 + y^2 = a^2$,then the locus of $M$ is