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# 0.2: Analytic geometry

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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

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
\label {conic}
ax^2+by^2+cx+dy+e = 0,

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 $$bc^2-ad^2-4abe=0.$$