There is a unique line that passes through
any two points \(A\bigl(x_1, y_2\bigr)\) and \(B\bigl(x_2,y_2\bigr).\)
The slope of that line is
\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2-y_1}{x_2-x_1}; \]
this is the “rate” at which the line is increasing/decreasing.
It’s also referred to as the pitch or grade.
The “point-slope” form, “slope-intercept” form,
and “general” form of a line are respectively are
\[ (y-y_1) = m(x-x_1) \qquad\qquad y = mx+b \qquad\qquad Ax + By = C \,. \]
Horizontal and vertical lines have special forms, \(y=b\) and \(x=a\) respectively.
Two lines are parallel if they never intersect, in which case they must have the same slope. Two lines are perpendicular (or normal, or orthogonal) if they intersect at a right angle, so one’s slope will be the negative reciprocal of the other. E.g. if a line has slope \(\frac{p}{q}\) another line perpendicular to that one will have slope \(-\frac{q}{p}.\)
The unit circle is the circle of radius 1 centered at the origin,
which corresponds with the equation \(x^2+y^2 = 1.\)
The line \(y = mx\) will intersect the top-half of the unit circle at the point
\[ (x,y) = \biggl(\frac{1}{\sqrt{1+m^2}}, \frac{m}{\sqrt{1+m^2}}\biggr)\,. \]