Coordinate Geometry

Distance Formula

The distance formula calculates the straight-line distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane.

\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Example:

Find the distance between points (3, 4) and (7, 1).

Solution:

\( d = \sqrt{(7-3)^2 + (1-4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \)

Slope of a Line

The slope of a line measures its steepness and is calculated as the ratio of the vertical change to the horizontal change between two points.

\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Example:

Find the slope of the line through points (2, 3) and (5, 11).

Solution:

\( m = \frac{11 - 3}{5 - 2} = \frac{8}{3} \)

Graphical Understanding of Slope

(2,3) (5,11) Base = 3 Height = 8

Possibilities Based on Intercepts

Depending on the slope and y-intercept, lines can appear in different positions on the coordinate plane:

  1. Positive slope, Positive y-intercept
  2. Positive slope, Negative y-intercept
  3. Negative slope, Positive y-intercept
  4. Negative slope, Negative y-intercept

Graphs of Common Functions

1. Parabola

A parabola is the set of all points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix).
\( \text{Vertical: } y = ax^2 + bx + c \quad \text{Horizontal: } x = ay^2 + by + c \)

2. Circle

A circle is the set of all points in a plane that are a fixed distance (radius) from a fixed point (center).
\( (x - h)^2 + (y - k)^2 = r^2 \)

3. Ellipse

An ellipse is the set of all points where the sum of the distances from two fixed points (foci) is constant.
\( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)

4. Rectangular Hyperbola

A rectangular hyperbola is a hyperbola whose asymptotes are perpendicular (at right angles).
\( xy = c^2 \)