Condition for Straight Lines in Plane to be Parallel/Slope Form/Proof 1
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Theorem
Let $L_1$ and $L_2$ be straight lines in the Cartesian plane.
Let the slope of $L_1$ and $L_2$ be $\mu_1$ and $\mu_2$ respectively.
Then $L_1$ is parallel to $L_2$ if and only if:
- $\mu_1 = \mu_2$
Proof
Let $L_1$ and $L_2$ be described by the general equation:
\(\ds L_1: \, \) | \(\ds l_1 x + m_1 y + n_1\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds L_2: \, \) | \(\ds l_2 x + m_2 y + n_2\) | \(=\) | \(\ds 0\) |
Then:
- the slope of $L_1$ is $\mu_1 = -\dfrac {l_1} {m_1}$
- the slope of $L_2$ is $\mu_2 = -\dfrac {l_2} {m_2}$.
From Condition for Straight Lines in Plane to be Parallel: General Equation:
- $L_1$ and $L_2$ are parallel if and only if $L_2$ is given by the equation:
- $m_1 x + m_2 y = n'$
- for some $n'$.
But then the slope of $L_2$ is $-\dfrac {l_1} {m_1}$.
That is:
- $-\dfrac {l_1} {m_1} = -\dfrac {l_2} {m_2}$
and the result follows.
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $5$