Math · 26 · Geometry and Trigonometry · Free

Circles in the xy-Plane

Notes

Circle: $(x-h)^2+(y-k)^2=r^2$ , centre $(h,k)$ , radius $r$ .
Signs flip: $(x+3)^2$ means $h=-3$ .
The right side is $r^2$ , not $r$ — take the square root.
Expanded form? Complete the square on $x$ and on $y$ separately.
A point is on the circle if it satisfies the equation exactly; inside if LHS < r².
Tangent line ⟂ radius at the point of contact.

Standard form

(x-h)^2+(y-k)^2=r^2

Completing the square

x^2+bx=\left(x+\tfrac{b}{2}\right)^2-\left(\tfrac{b}{2}\right)^2

Distance between points

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

Midpoint (centre from a diameter)

\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)

On Desmos, type the equation and click the centre — it reads off the coordinates and radius for you.

10 official SAT questions on Circles in the xy-Plane, easiest first.