The equation of the base of an equilateral triangle is $12x+5y-65=0$. If one of its vertices is $(2,3)$,then the length of the side is

What is the position of the point $(8, -9)$ with respect to the lines $2x + 3y - 4 = 0$ and $6x + 9y + 8 = 0$?

The angle made by a line $L$ with the positive $X$-axis measured in the positive direction is $\frac{\pi}{6}$ and the intercept made by $L$ on the $Y$-axis is negative. If $L$ is at a distance of $5$ units from the origin,then the perpendicular distance from the point $(1, -\sqrt{3})$ to the line $L$ is

If ${p_1}, {p_2}$ and ${p_3}$ are the perpendicular distances from the points $({m^2}, 2m)$,$(mm', m + m')$ and $(m'^2, 2m')$ respectively to the line $x \cos \alpha + y \sin \alpha + \frac{\sin^2 \alpha}{\cos \alpha} = 0$,then ${p_1}, {p_2}$ and ${p_3}$ are in:

Let the lines be $L_1: 2x + 3y - 7 = 0$ and $L_2: 2x + 3y - 12 = 0$. For the point $A(3, -5)$,which of the following is true?

The perpendicular distance from the point $(1, \pi)$ to the line joining $(1, 0^{\circ})$ and $(1, \frac{\pi}{2})$ (in polar coordinates) is