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

The equation of the line whose slope is $3$ and which cuts off an intercept of $3$ from the positive $x$-axis is

The equation of a line,whose perpendicular distance from the origin is $7$ units and the angle,which the perpendicular to the line from the origin makes,is $120^{\circ}$ with the positive $X$-axis,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 the length of the perpendicular to a line from the origin is $2 \sqrt{2}$ units,which makes an angle of $135^{\circ}$ with the $X$-axis,then the equation of the line is

Find the equation of the line which intersects the $y$-axis at a distance of $2$ units above the origin and makes an angle of $30^{\circ}$ with the positive direction of the $x$-axis.