0.2: Analytic geometry

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$Thangarajah.jpg$
Contributed by Pamini Thangarajah
Professor (Mathematics & Computing) at Mount Royal University

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The straight line

Theorem $\PageIndex{1}$

Given two points $A=(x_1,y_1)$ and $B=(x_2,y_2),$

1. The slope of $AB\,\,=\displaystyle \frac{y_2-y_1}{x_2-x_1}=\displaystyle \frac{\mbox{ rise}}{\mbox{ run}}.$

2. The length of the line segment $AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.$

3. The mid-point $C,$ of the line $AB$ has coordinates $\left(\displaystyle \frac{(x_1+x_2)}{2},
\displaystyle \frac{(y_1+y_2)}{2}\right).$

It is often useful to be able to describe the path or locus of a point as a relationship between the coordinates of an arbitary point on the path. Some of the standard paths are described below. It is worth noting here that the equation of a straight line is completely determined if we know the slope of the line and one point on the line.

Equations of the straight line

1. The point slope form

The equation of the straight line which has slope $m$ and which passes through the point $(x_1,y_1)$ is, $${y-y_1=m(x-x_1)}.$$

2. The standard form

The equation of the straight line which has slope $m$ and makes an intercept of $b$ units on the $y-$axis is $${y=mx+b}.$$

Any straight line in the plane has an equation of the form $${ ax+by+c=0}.$$
Where $a,b$ and $c$ are real numbers. This straight line has slope $ \displaystyle \frac {-a}{b}$ and $y-$intercept $\displaystyle \frac{-c}{b},$ provided $b\ne 0.$

Parallel straight lines and perpendicular straight lines

Slopes

1. Two straight lines are parallel if and only if their slopes are equal.
If $L_1$ is a straight line with slope $m_1$ and $L_2$ is a straight line with slope $m_2$, and if neither line is parallel to either $x-$axis or the $y-$axis, then $L_1$ is perpendicular to $L_2$ if and only if $m_1m_2=-1.$

(Alternatively we note that the slope of $L_1$ is the negative reciprocal of slope of $L_2$.)

2. The perpendicular distance of a point $(h,k)$ from the straight line with equation $ax+by+c=0$ is given by
$$d=\frac{|ah+bk+c|}{\sqrt{a^2+b^2}}.$$
Though it is not essential to memorize this formula, it can sometimes be very useful.

The circle

Circle

1. The equation of a circle with the centre $(h,k)$ and the radius $r$ is $$(x-h)^2+(y-k)^2=r^2.$$
2. The general form of the equation of a circle is $$x^2+y^2+2gx+2fy+c=0,$$ this circle has centre $(-g,-f)$ and radius $\sqrt{g^2+f^2-c}.$

The conic sections

Conic Sections

The general form of the equation of the conic section which has axes parrallel to the $x-$ and $y-$ axes is
\begin{equation}\label {conic}
ax^2+by^2+cx+dy+e = 0,
\end{equation}
provided that $a$ and $b$ are not both zero.

The conic sections may be readily identified using the information given below:
Equation ~\ref{conic} represents

1. a circle provided that $a=b$.
2. a parabola with axis parallel to the $x-$axis if $a=0$, and $b \ne 0.$
3. a parabola with axis parallel to the $y-$axis if $b=0$, and $a \ne 0.$
4. an ellipse provided that $a \ne b$ and both $a$ and $b$ have the same sign.
5. a hyperbola provided that $a$ and $b$ differ in sign.
6. two straight lines exactly when $bc^2-ad^2-4abe=0.$