Straight Line
The general form of the equation is: $Ax + By + C = 0$
The standard form of the equation is $y = mx + C$ this also known as slope-intercept form
The relative change, $\frac{\Delta y}{\Delta x}$ is the gradient of the line segment.
The point-slope form is $y - y_1 = m(x - x_1)$
The angle between 2 lines which has slope $m_1$ and $m_2$:
$$\tan \theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$$
The distance between two points is:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
If two lines are parallel then –> $m_1 = m_2$
If two lines are Perpendicular then –> $m_1 \cdot m_2 = -1$
Quadartic Equation
The general form of equation is: $$ax^2+bx+c = 0$$
The relationship between $x$ and $y$ is non-linear and symmetrical
The most fundamental reason this quadartic equation is a parabola because of is geometric definitions
A parabola is set of all points in a plane that are an equal distance from a specific point (the focus) and a specific line (the directrix).
Parabola
| Form | Equation |
|---|---|
| General | $f(x) = Ax^2 + Bx + C$ |
| Standard $(y-k)^2 = 2p(x-h)$ | |
| Standard $y^2= 2px$ at vertex (0,0) or h = k = 0 –> $Focus (p/2,0)$ ; Equation of directrix : $x = -p/2$ |
The eccentricity of parabola = 1
Ellipse
The general form of conic section $Ax^2+Bxy+Cy^2+Dx+Ey+F = 0$
- if $B^2-4AC < 0$ an Ellipse is defined
- if $B^2-4AC > 0$ an Hyperbola is defined
- if $B^2-4AC = 0$ an Parabola is defined
- if $B^2-4AC < 0$ an Ellipse is defined
- if $A=C$ and $B= 0$ an Circle is defined
- if $A=B=C=0$ an Straight line is defined
The Ellipse (Major is x axis)
Circle
The standard form equation of circle with center $(a,b)$ is $(x-a)^2 + (y-b)^2 = c^2$
The equation of circle can be written in $x^2+y^2+2gx+2fy+c =0$ $where$ $center$ $is$ $(-g,-f)$ and $radius = \sqrt{g^2+f^2-c}$
