If the straight line through the point $P(3, 4)$ makes an angle $\frac{\pi}{6}$ with the $x$-axis and meets the line $12x + 5y + 10 = 0$ at $Q$,then the length $PQ$ is

The equation of the line which makes a right-angled triangle with the axes whose area is $6$ sq. units and whose hypotenuse is $5$ units,is

If a line $L$ passes through the point of intersection of the lines $2x + 3y + 1 = 0$ and $x + y - 3 = 0$ and the angle made by the line $L$ with the $X$-axis measured in the positive direction from the positive $X$-axis is $\tan^{-1} \frac{2}{3}$,then the sum of the intercepts made by this line on the axes is

The equation of the line passing through the point $(-3, 2)$ and parallel to the $x$-axis is:

Find the equation of the line passing through the point $(2, 3)$ and perpendicular to the line joining the points $(-5, 6)$ and $(-6, 5)$.

The line $PQ$ with equation $x - y = 2$ intersects the $x$-axis at $P$. Given $Q$ is $(4, 2)$. Find the equation of the line $PQ$ after it is rotated by $45^{\circ}$ in the counter-clockwise direction about point $P$.