Find the length of the perpendicular from the point (x1, y1) to the straight line Ax + By + C = 0, the axes being inclined at an angle ω, and the equation being written such that C is a negative quantity.
Ax1+By1+C√A2+B2−2AB cos ω.sin ω
Let the given straight line meet the axis in L and M. So that OL=−CA and OM=−CB
Let P be the given point (x1, y1). Draw the perpendicular PQ, PR and PS on the given line and two axes.
Taking O and P on opposite sides of the given line, we then have
ΔLPM+ΔMOL=ΔOLP+ΔOPM
i.e. PQ.LM + OL. OM sin ω = OL.PR + OM. PS - - - (1)
Draw PU and PV parallel to the axes of y and x, such that PU=y1 and PV = x1
Hence, PR = PU sin PUR =y1 sin ω
and PS = PV sin PVS = x1 sin ω
Also, LM=√OL2+OM2−2OL.OM cos ω=√C2A2+C2B2−2C2ABcos ω=−C√1A2+1B2−2 cos ωAB
Since, C is a negative quantity
On substituting these values in (1), we have
PQ×(−C)×√1A2+1B2−2 cos ωAB+C2ABsin ω=−CAy1 sin ω−CBx1 sin ω
On simplifying,
PQ=Ax1+By1+C√A2+B2−2AB cos ω.sin ω