Find the equation of the line that bisects the line segment joining the points $(5, 3)$ and $(4, 4)$ and makes an angle of $45^{\circ}$ with the positive direction of the $x$-axis.

Find the equation of a line that makes an intercept of $-3$ on the $y$-axis and makes an angle of ${\tan ^{ - 1}}\frac{3}{5}$ with the $x$-axis.

Let $u = \hat{i} - 2\hat{j}$ and $v = -3\hat{i} + 5\hat{j}$. Consider three points $P, Q,$ and $R$ having position vectors $-\frac{1}{7}\hat{i}$,$-\frac{1}{4}\hat{j}$,and $-2\hat{i} + 3\hat{j}$ respectively. Among these,the points on the line segment passing through $u$ and $v$ are

In what ratio is the line segment joining the points $(-1, 1)$ and $(5, 7)$ divided by the line $x + y = 4$?

The equation of the line passing through $(1, 2)$ and perpendicular to $3x + 4y + 5 = 0$ is

The points on the straight line $3x - 4y + 1 = 0$ which are at a distance of $5$ units from the point $(3, 2)$ are