Find the equation of the line perpendicular to the line $2x - 3y = 5$ and passing through the point $(1, -1)$.

$A$ line through the point $A(2,0)$ which makes an angle of $30^{\circ}$ with the positive direction of $x$-axis is rotated about $A$ in clockwise direction through an angle $15^{\circ}$. Then the equation of the straight line in the new position is

Find the value of $x$ for which the points $(x, -1), (2, 1),$ and $(4, 5)$ are collinear.

The angles made with the $X$-axis by the two lines passing through the point $P(1, 2)$ and cutting the line $x+y=4$ at a distance $\frac{\sqrt{6}}{3}$ units from the point $P$ are

If the line $2x + 3y = 5$ and $y = mx + c$ are parallel,then:

If a line $L$ passing through the point $A(-2, 4)$ makes an angle of $60^{\circ}$ with the positive direction of $X$-axis in the anti-clockwise direction and $B(p, q)$ lying in the $3^{\text{rd}}$ quadrant is a point on $L$ at a distance of $6$ units from the point $A$,then $\sqrt{p^2+q^2-8q} = $