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

Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point $(2,3)$.

$A$ line is passing through the point $(4,3)$ and the sum of its intercepts made on the coordinate axes is $14$. Then an equation of that line is

In the equation $y - y_1 = m(x - x_1)$,if $m$ and $x_1$ are fixed and different lines are drawn for different values of $y_1$,then

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 $\frac{a^2}{p^2}+\frac{b^2}{q^2}=$

Let the coordinates of the two points $A$ and $B$ be $(1, 2)$ and $(7, 5)$ respectively. The line $AB$ is rotated through $45^{\circ}$ in the anti-clockwise direction about the point of trisection of $AB$ which is nearer to $B$. The equation of the line in the new position is: