Coordinate Geometry: Equations of Lines and Curves, Parametric Equations

1. Equations of Lines

1.1 Slope-Intercept Form

The equation of a straight line can be expressed in slope-intercept form as:

$y = mx + c$

where:

• $m$ is the gradient (slope) of the line.
• $c$ is the y-intercept (the point where the line crosses the y-axis).

To find the gradient, use the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

where $(x_1, y_1)$ and $(x_2, y_2)$ are two distinct points on the line.

1.2 Point-Slope Form

Given a point $(x_1, y_1)$ on the line and a gradient $m$, the point-slope form of the equation is:

$y - y_1 = m(x - x_1)$

This form is useful for finding the equation of a line when you know a point on the line and its slope.

1.3 General Form

The general form of the equation of a line is:

$Ax + By + C = 0$

where $A$, $B$, and $C$ are constants.

From the general form, we can find the gradient $m$ by rearranging the equation:

$y = -\frac{A}{B}x - \frac{C}{B}$

So, $m = -\frac{A}{B}$

1.4 Parallel and Perpendicular Lines

Parallel Lines: Two lines are parallel if and only if they have the same gradient.

Perpendicular Lines: Two lines are perpendicular if and only if the product of their gradients is -1. If the gradient of one line is $m$, the gradient of a perpendicular line is $-\frac{1}{m}$.

1.5 Intercepts

x-intercept: The point where the line crosses the x-axis (where $y = 0$). To find the x-intercept, set $y = 0$ in the equation of the line and solve for $x$.

y-intercept: The point where the line crosses the y-axis (where $x = 0$). To find the y-intercept, set $x = 0$ in the equation of the line and solve for $y$.

2. Equations of Curves

2.1 Circle

The equation of a circle with centre $(a, b)$ and radius $r$ is:

$(x - a)^2 + (y - b)^2 = r^2$

2.2 Parabola

The standard forms of the equations of parabolas are:

• Vertical Parabola: $y = ax^2 + bx + c$
• Horizontal Parabola: $x = ay^2 + by + c$

2.3 Ellipse

The standard form of the equation of an ellipse centred at the origin is:

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

where $a$ is the semi-major axis and $b$ is the semi-minor axis.

2.4 Hyperbola

The standard forms of the equations of hyperbolas centred at the origin are:

• Horizontal Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
• Vertical Hyperbola: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$

3. Parametric Equations

3.1 Definition

Parametric equations define the coordinates of a point $(x, y)$ in terms of a single parameter, usually denoted by $t$.

$x = f(t)$ and $y = g(t)$

As the parameter $t$ varies, the point $(x, y)$ traces out a curve.

3.2 Eliminating the Parameter

Sometimes, it's necessary to eliminate the parameter $t$ to find the Cartesian equation of the curve.

To do this, solve one equation for $t$ and substitute the expression for $t$ into the other equation.

3.3 Examples

1. Circle: For a circle with centre $(a, b)$ and radius $r$, the parametric equations are:

   $x = a + r\cos(t)$

   $y = b + r\sin(t)$

2. Parabola: For a vertical parabola $y = ax^2 + bx + c$, the parametric equations are:

   $x = t$

   $y = at^2 + bt + c$

Topic	Key Concepts
Equations of Lines	Slope-intercept form, point-slope form, general form, parallel and perpendicular lines, intercepts
Equations of Curves	Circle, parabola, ellipse, hyperbola – standard forms
Parametric Equations	Definition, eliminating the parameter, examples for circle and parabola