[b]uEquations of circles[/u]/b
There are 2 forms for the equation of a circle.
uThe "completed square" form/u
row
col((x-a)2+(y-b)2=r2)/col
colwhere (r) is the radius and ((a,b)) is the centre of the circle.[/col]/row
Multiply out both brackets and rearrange to convert to the "multiplied out" form.
uThe "multiplied out" form/u
row
col(x2+y2-2px-2qy+s=0)/col
colwhere (\sqrt{p2+q2-s}) is the radius and ((p,q)) is the centre of the circle.[/col]/row
Complete the square for both (x) and (y) to convert to the "completed square" form.
[b]uStraight lines and circles[/u]/b
A straight line can intersect a circle once, twice, or not at all.
If a straight line intersects a circle once, by touching it, then it also a tangent of the circle.
row
[col]mtaimg/images/topics/3/3-28-1.png[/mtaimg]pIntersect once (tangent)[/p]/col
[col]mtaimg/images/topics/3/3-28-2.png[/mtaimg]pIntersect twice[/p]/col
[col]mtaimg/images/topics/3/3-28-3.png[/mtaimg]pDoes not intersect[/p][/col]/row
[b]uProperties of circles[/u]/b
[row][col]mtaimg/images/topics/3/3-28-4.png[/mtaimg]pA tangent to a circle is perpendicular to the radius of the circle at the point of intersection.[/p]/col
[col]mtaimg/images/topics/3/3-28-5.png[/mtaimg]pThe perpendicular bisector of a chord will go through the centre of a circle.[/p]/col
[col]mtaimg/images/topics/3/3-28-6.png[/mtaimg]pThe angle in a semicircle is a right angle.[/p][/col]/row
With these properties, the equation of a circumcircle of a triangle with given vertices and the equation of a tangent at a specified point on the circumference can be found.
There are 2 forms for the equation of a circle:
ul
li((x-a)2+(y-b)2=r2), radius is (r) and center is ((a,b))/li
li(x2+y2-2px-2qy+s=0), radius is (\sqrt{p^2+q^2-s}) and center is ((p,q))[/li]/ul