Equations of Circles

Circles (Lesson Objectives) Write the standard form of the equation of a circle. Graph a circle by hand using the standard form of the equation of a circle. Write the general form of an equation of a circle using center and radius and identify center and radius from the general form of equation of a circle.

Standard Form of a Circle Center is at (h, k)and radius r r is the radius of the circle

EX 1 Write the standard form for the equation of a circle with center (3, -2) and a radius of 4. Draw it. h k r

EX 2 Write an equation of a circle with center (-4, 0) and a diameter of 10. k

EX 3 Write an equation of a circle with center (2, -9) and a radius of . k r

Opposite signs! ( , ) 6 -3 Take the square root! Radius 25 EX 4 Find the coordinates of the center and the measure of the radius. Opposite signs! ( , ) 6 -3 Take the square root! Radius 25

5. Find the center, radius, & equation of the circle. The center is The radius is The equation is (0, 0) 12 x2 + y2 = 144

7. Graph the circle, identify the center & radius. (x – 3)2 + (y – 2)2 = 9 Center (3, 2) Radius of 3

General Form of a Circle General form of equation of a circle is X2 + Y2+2gx+2fy+c = 0 In this equation center is (-g, -f) Radius is √f2+g2 – c Also in general form of a circle, the coefficients of x² and y² has to be same, otherwise it is not a circle equation. To calculate center and radius, make sure you reduce the coefficients to 1 by dividing by the coefficients of x² and y²

Example: What is center and radius of x²+y²+4x+6y-2 = 0 2g = 4 g=2 2f=6 f=3 In this problem g = 2 f = 3 and c = -2 So center is (-2, -3) = (-2, -3) and radius = √(2)²+(3)²-(-2)  = √13+2 = √15 Another example x²+y²-8x-4y-5=0 2g=-8 g=-4 2f=-4 f=-2 c=-5 Center = (-(-4), -(-2)) = (4, 2) and radius = √(-4)²+(-2)²-(-5) = √25 = 5

An Example on general form with coefficients not equal to 1 4x²+4y²-36x-8y-48=0 Divide by 4 on both sides x²+y²-9x-2y-12 = 0 2g=-9 2f=-2 g=-9/2 and f= -1 and c=-12 now center is (9/2, 1)  Radius = √(-9/2)²+(-1)²-(-12) = √33.25 = 5.77

Another example Determine equation of a circle with center at (0,1) and which passes through (2,3) Radius = Distance from center to point (2,3) =√(0-2)²+(1-3)² = √8 So equation is (x-0)²+(y-1)²=(√8)²                        x²+(y-1)²=8 We can convert it to general form too as follows                        x²+(y-1)(y-1) = 8                         x²+y²-2y+1=8                          x²+y²-2y-7=0