# How do you find the length of the line x=At+B, y=Ct+D, a<=t<=b?

Mar 4, 2018

$l = \left\mid a - b \right\mid \sqrt{{A}^{2} + {C}^{2}}$

#### Explanation:

Given the line

$L \to p = {p}_{0} + \vec{v} t$

with

$p = \left(x , y\right)$
${p}_{0} = \left(B , D\right)$
$\vec{v} = \left(A , C\right)$

we have

${p}_{a} = {p}_{0} + \vec{v} a$
${p}_{b} = {p}_{0} + \vec{v} b$

and then

$l = \left\lVert {p}_{a} - {p}_{b} \right\rVert = \left\lVert {p}_{0} + \left(\vec{v}\right) a - {p}_{0} - \left(\vec{v}\right) b \right\rVert = \left\lVert \vec{v} \left(a - b\right) \right\rVert = \left\mid a - b \right\mid \left\lVert \vec{v} \right\rVert = \left\mid a - b \right\mid \sqrt{{A}^{2} + {C}^{2}}$