The equation of the line perpendicular to $x = c$ is

If a straight line passing 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 of $PQ$ is

The straight line passing through the point of intersection of the straight lines $x - 3y + 1 = 0$ and $2x + 5y - 9 = 0$ and having infinite slope and at a distance of $2 \text{ units}$ from the origin,has the equation

If the coordinates of the points $A, B, C, D$ are $(a, b), (a', b'), (-a, b)$ and $(a', -b')$ respectively,then the equation of the line bisecting the line segments $AB$ and $CD$ is

Prove that the equation of the line passing through the point $(x_{1}, y_{1})$ and parallel to the line $Ax + By + C = 0$ is $A(x - x_{1}) + B(y - y_{1}) = 0$.

If a straight line through the point $P(1, 2)$,which makes an angle $45^{\circ}$ with the $X$-axis,meets the line $3x + 4y + 5 = 0$ at $Q$,then the length of $PQ$ equals ......... units.