Theory

Analytic geometry and conic sections

Points and lines in the Cartesian plane, then parabola, circle, ellipse and hyperbola, with the method for recognizing a conic from its equation and drawing it.

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The Cartesian plane
The line

The Cartesian plane

Distance between two points and midpoint

Given $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ :

\overline{A B} = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}, M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}) .

The distance is the Pythagorean theorem applied to the right triangle with legs $∣ x_{2} - x_{1} ∣$ and $∣ y_{2} - y_{1} ∣$ ; the midpoint has as coordinates the averages of the coordinates. These two formulas are used throughout the chapter: the circle, the ellipse and the hyperbola are defined precisely in terms of distances.

Example

$A (1, 2)$ and $B (4, 6)$ : the distance is $\overline{A B} = 3^{2} + 4^{2} = 25 = 5$ and the midpoint is $M (\frac{5}{2}, 4)$ .

The line

1. Equation of a line

A non-vertical line has slope-intercept equation

y = m x + q,

where $m$ is the slope and $q$ is the ordinate of the point of intersection with the $y$ -axis. A vertical line has equation $x = c$ . Every line, vertical or not, can be written in general form

a x + b y + c = 0, (a, b) \neq = (0, 0);

for $b \neq = 0$ the two forms translate into each other through $m = - \frac{a}{b}$ and $q = - \frac{c}{b}$ . The slope-intercept form is read as a function and can be drawn immediately; the general form also includes vertical lines and appears in the point-line distance formula.

2. Slope, lines through a point, parallelism and perpendicularity

The slope of the line through two points $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ with $x_{1} \neq = x_{2}$ is

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}},

the ratio between the change in $y$ and the change in $x$ . The line with slope $m$ passing through the point $P (x_{0}, y_{0})$ is

y - y_{0} = m (x - x_{0}) .

Two non-vertical lines with slopes $m_{1}$ and $m_{2}$ are parallel if and only if $m_{1} = m_{2}$ ; they are perpendicular if and only if $m_{1} m_{2} = - 1$ , that is $m_{2} = - \frac{1}{m _{1}}$ . With these three formulas one writes the required line in almost every exercise: one finds $m$ from the data and imposes the passage through the point.

Example

The line through $P (1, 3)$ perpendicular to $y = 2 x - 5$ has slope $m = - \frac{1}{2}$ , therefore it is $y - 3 = - \frac{1}{2} (x - 1)$ , that is $y = - \frac{1}{2} x + \frac{7}{2}$ .

3. Distance from a point to a line

The distance from the point $P (x_{0}, y_{0})$ to the line $r$ of equation $a x + b y + c = 0$ is

d (P, r) = \frac{∣ a x _{0} + b y _{0} + c ∣}{a ^{2} + b ^{2}} .

The line must be written in general form before applying the formula. The point-line distance is also used with conic sections, for example to impose that a line be tangent to a circle: tangency is equivalent to the equality between the distance from the center to the line and the radius.

Example

Distance from $P (2, 1)$ to the line $3 x - 4 y + 1 = 0$ :

d = \frac{∣3 \cdot 2 - 4 \cdot 1 + 1∣}{9 + 16} = \frac{3}{5} .

The theory continues with 5 more sections (the parabola, the circle, the ellipse…), plus all worked examples, the solutions and the PDF.

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