# 0.2: Analytic geometry

- Page ID
- 10822

## 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

\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 \(bc^2-ad^2-4abe=0.\)