The equation of a line passing through the point $(7, -4)$ and perpendicular to the line passing through the points $(2, 3)$ and $(1, -2)$ is

The points $A (1, 3)$ and $C (5, 1)$ are the opposite vertices of a rectangle. The equation of the line passing through the other two vertices and having a gradient of $2$ is:

The equation of the line passing through $(c, d)$ and parallel to $ax + by + c = 0$ is

If $p$ and $q$ are the $x$ and $y$-intercepts respectively of the line passing through the points $(a \cos \alpha, b \sin \alpha)$ and $(a \cos \beta, b \sin \beta)$,then $\frac{a^2}{p^2}+\frac{b^2}{q^2}=$

The line joining two points $A(2, 0)$ and $B(3, 1)$ is rotated about $A$ in the anticlockwise direction through an angle of $15^{\circ}$. The equation of the line in the new position is:

$A$ line $L$ passes through the point $P(1, 2)$ and makes an angle of $60^{\circ}$ with the positive $X$-axis. $A$ and $B$ are two points lying on $L$ at a distance of $4$ units from $P$. If $O$ is the origin,then the area of $\triangle OAB$ is