# Length of Perpendicular from a Point to a Line

**\(\textbf{Art 7 : } \qquad\boxed{{\text{Length of perpendicular}}}\)**

Suppose that we are given the equation of a line *L* and we are required to find the length of the perpendicular dropped from an arbitrary point \(P\left( {{x_1},{y_1}} \right)\,\,{\text{on}}\,\,L.\)

Suppose that the equation of *L* is in normal form, i.e,\(L \equiv x\cos \alpha + y\sin \alpha = p.\)

Based on the discussion in the figure above, the equation of the line \({L^{\,{\mathbf{'}}}}\,\,{\text{is}}\,\,x\cos \alpha + y\sin \alpha - {p_1} = 0.\) Since *L*_{1} passes through *P*, the co-ordinates of *P* must satisfy the equation of *L*_{1}. Thus,

\[{x_1}\cos \alpha + {y_1}\sin \alpha - {p_1} = 0\]

Thus, we get \({p_1}\) as \(\left( {{x_1}\cos \alpha + {y_1}\sin \alpha } \right).\) The length of perpendicular *PQ* is now simply \(\left| {{p_1} - p} \right| = \left| {{x_1}\cos \alpha + {y_1}\sin \alpha - p} \right|\). (Modulus sign is used since *PQ* is a length so it must be positive).

\[\boxed{PQ = \left| {{x_1}\cos \alpha + {y_1}\sin \alpha - p} \right|}\,\,\,\,:\,\,\,\,{\mathbf{Length}}{\text{ }}{\mathbf{of}}{\text{ }}{\mathbf{perpendicular}}\]

Let us now assume the case where *L* is given in the general form, i.e.\(L \equiv ax + by + c = 0.\)

We can easily adjust the equation of *L* so that *c* is negative. We do this so that we can convert *L* into the normal form:

\[\begin{align}& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\qquad ax + by + c = 0\qquad c < {\text{ }}0 \\&\qquad \Rightarrow \quad ax + by = - c \\&\qquad\Rightarrow \quad \left( {\frac{a}{{\sqrt {{a^2} + {b^2}} }}} \right)x + \left( {\frac{b}{{\sqrt {{a^2} + {b^2}} }}} \right)y = \left( {\frac{{ - c}}{{\sqrt {{a^2} + {b^2}} }}} \right) \\&\qquad

\Rightarrow \quad x\cos \alpha + y\sin \alpha = p\qquad \qquad \qquad \qquad \qquad \qquad \qquad ...{\text{ }}\left( 1 \right) \\& {\text{where}}\;\;\;\cos \alpha = \frac{a}{{\sqrt {{a^2} + {b^2}} }},\,\,\sin \alpha = \frac{b}{{\sqrt {{a^2} + {b^2}} }}\,\,{\text{and}}\,\,p = \frac{{ - c}}{{\sqrt {{a^2} + {b^2}} }} \end{align} \]

The equation in (1) is in the normal form; we can now use the result obtained in the preceding discussion to obtain the length of the perpendicular *PQ*:

\[\begin{align}& PQ = \left| {{x_1}\cos \alpha + {y_1}\sin \alpha - p} \right|\,\,\,\,\left\{ \begin{gathered}

{\text{Modulus sign is}} \\ {\text{used since }}PQ \\ {\text{must be + ve}} \end{gathered} \right\} \\

& \qquad = \left| {\frac{{a{x_1}}}{{\sqrt {{a^2} + {b^2}} }} + \frac{{b{y_1}}}{{\sqrt {{a^2} + {b^2}} }} + \frac{c}{{\sqrt {{a^2} + {b^2}} }}} \right| \\& \boxed{PQ = \frac{{\left| {a{x_1} + b{y_1} + c} \right|}}{{\sqrt {{a^2} + {b^2}} }}}\qquad :\qquad {\mathbf{Length}}{\text{ }}{\mathbf{of}}{\text{ }}{\mathbf{perpendicular}} \end{align} \]

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