Foot, Image ,Distance and Angle

Foot, Image ,Distance and Angle
Distances 
(1) The perpendicular distance to the line ax+by+c=0
a) from origin is \(\frac{{|c|}}{{\sqrt {{a^2} + {b^2}} }}\)
b) from the point \((x_1,y_1)\) is \(\frac{{|a{x_1} + b{y_1} + c|}}{{\sqrt {{a^2} + {b^2}} }}\)
(ii) The distance of a point \((x_1,y_1)\) from The line \(L \equiv \) ax + by +c=0 measured along a line making an angle \(\alpha \) with x-axis is \(|\frac{{a{x_1} + b{y_1} + c}}{{a\cos \alpha + b\sin \alpha }}|\)
(iii)the distance between the parallel lines \(ax + by + {c_1}\) and \(ax + by + {c_2}\) =0 is \(\frac{{|{c_1} - {c_2}|}}{{\sqrt {{a^2} + {b^2}} }}\)

(iv) The distance between the parallel  lines \(ax + by + {c_1}\) and \(ax + by + {c_2}\) measured along the line having inclination \(\theta\) is \(|\frac{{{c_1} - {c_2}}}{{a\cos \theta + b\sin \theta }}|\)