Lines

Samia CHALLAL

24 Lines

The slope $m$ of a non vertical line that passes through the points $A(x_A, y_A)$ and $B(x_B, y_B)$ is

$\quad \displaystyle{m=\frac{y_B-y_A}{x_B-x_A} }$

The slope of a line is independent on the choice of two points on the line.

$\bullet$ A vertical line has an infinite slope (or no slope).

$\bullet$ An equation of the line that passes through the point $(x_0, y_0)$ and has slope $m$ is

$\quad y-y_0 = m (x-x_0).$

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$\bullet$ Two non vertical lines are parallel “// ” if and only if they have the same slope.

$\bullet$ Two non vertical lines are perpendicular ” $\bot$ ” if and only if their slope are negative reciprocals.
If $L_1 : y=m_1 x + b_1 , \quad L_2 : y=m_2 x + b_2,$ then

$\qquad L_1\,//\, L_2 \qquad \Longleftrightarrow \qquad m_1=m_2$

$\qquad L_1\, \bot\, L_2 \qquad \Longleftrightarrow \qquad m_1. m_2 = -1$

$\bullet$ A horizontal line (slope 0) is perpendicular to a vertical line (no slope)

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Exercise 1

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The line passes through the points $(0,-5)$ and $(2,-1)$ .

The general equation of this line is in the form : $y= m x + b$

For $x=0$ , we have $y=-5 =b$ .

For $y=-1$ , we obtain $-1= m (2) + b$ .

Thus $b=-5$ and $m=2$ .

The equation of the line is : $y= 2 x -5$ .

Exercise 2

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Line $L_4$ passes through the points $(0,4)$ and $(-2,0)$ . Thus its equation is $y=2 x + 4$ .

Line $L_3$ is horizontal and passes through the points $(0,-3)$ . Thus its slope is zero and its equation is : $y= -3$

Line $L_2$ passes through the points $(0,-3)$ and $(2,0)$ . Thus its equation is $\displaystyle{y=\frac{3}{2} x - 3}$ .

Finally, the equation of line $L_1$ is : $\displaystyle{y= -\frac{2}{3} x + 4 }$ .

Exercise 3

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The equation of the line that passes through the points $(6,0)$ and $(3,-2)$ is given by

$\displaystyle{ \frac{y-0}{(x-(6))}= \frac{-2-0}{3-(6)} \quad \Longleftrightarrow\quad y= \frac{2}{3} (x-(6)) }$

The equation of the line that passes through the points $(0,0)$ and $(3,2)$ is given by

$\displaystyle{ \frac{y-0}{(x-(0))}= \frac{2-0}{3-(0)} \quad \Longleftrightarrow\quad y= \frac{2}{3} x }$

The two lines are parallel since they have the same slope $2/3$ .

Exercise 4

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The equation of the line that passes through the points $(0,0)$ and $(6,4)$ is given by

$\displaystyle{ \frac{y-0}{(x-0)}= \frac{4-0}{6-0} \quad \Longleftrightarrow\quad y= \frac{2}{3} x }$

The equation of the line that passes through the points $(4,0)$ and $(0,6)$ is given by

$\displaystyle{ \frac{y-0}{(x-4)}= \frac{6-0}{0-4} \quad \Longleftrightarrow\quad y= -\frac{3}{2} (x - 4)}$

The two lines are perpendicular since their slopes are negative reciprocals : $\displaystyle{ - \frac{3}{2} . \frac{2}{3} = -1}$ .

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