08 Analytic Geometry Free Online FlashCards

What signs identify Quadrant II?

Quadrant II has negative $x$-coordinates and positive $y$-coordinates.

How are the two coordinate intercepts defined?

An $x$-intercept has $y=0$; a $y$-intercept has $x=0$.

What is the distance formula between two points?

For $(x_1,y_1)$ and $(x_2,y_2)$, the distance is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.

What is the distance from $(-3,-1)$ to $(2,3)$?

For $A(-3,-1)$ and $B(2,3)$, the distance is $\sqrt{41}$ units, approximately $6.40$ units.

What is the midpoint formula?

The midpoint is $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the two $x$-coordinates and the two $y$-coordinates.

How can an unknown endpoint be found from a midpoint?

Use $x_2=2h-x_1$ and $y_2=2k-y_1$. These equations recover the unknown endpoint from midpoint $(h,k)$ and endpoint $(x_1,y_1)$.

What does the slope formula measure?

The slope is $m=\frac{y_2-y_1}{x_2-x_1}$, provided $x_2\ne x_1$. It measures vertical change relative to horizontal change.

How do vertical and horizontal lines differ in slope?

A vertical line has equation $x=c$ and undefined slope; a horizontal line has equation $y=c$ and slope $0$.

What are three common forms of a line equation?

Slope-intercept form is $y=mx+b$; point-slope form is $y-y_1=m(x-x_1)$; standard form is $Ax+By=C$.

How are parallel and perpendicular lines identified by slope?

Parallel nonvertical lines have equal slopes. Perpendicular nonvertical lines have slopes whose product is $-1$.

How can the intersection of two lines be found?

Set the two expressions for $y$ equal, solve for $x$, and substitute that value into either equation to find $y$.

What is the standard equation of a circle?

A circle with center $(h,k)$ and radius $r$ has equation $(x-h)^2+(y-k)^2=r^2$.