Overview
This lecture explains how to convert between the standard and general forms of the equation of a circle, and how to find the circle's center and radius from these forms.
General and Standard Forms of a Circle
- The general form of a circle: ( x^2 + y^2 + Dx + Ey + F = 0 )
- The standard form of a circle: ( (x - h)^2 + (y - k)^2 = r^2 ), where (h, k) is the center and r is the radius.
- The general form is derived by expanding and arranging the standard form.
- To convert from standard to general: expand binomials and combine like terms.
Converting Standard to General Form (Example)
- Given center (4, -1), radius 7: ( (x-4)^2 + (y+1)^2 = 49 )
- Expanding: ( x^2 + y^2 - 8x + 2y - 32 = 0 )
- Steps: expand binomials, combine like terms, set equal to zero.
Converting General to Standard Form
- Use completing the square for x and y terms in the general form.
- Rearrange terms: group x's and y's, move constant to the other side.
- Complete the square for x: halve middle term, square result, add on both sides.
- Repeat for y, then write in standard form.
Examples of Completing the Square
- ( x^2 + y^2 + 6x - 7 = 0 ) becomes ( (x + 3)^2 + y^2 = 16 )
- ( x^2 + y^2 - 4x + 18y + 35 = 0 ) becomes ( (x - 2)^2 + (y + 9)^2 = 50 )
- ( x^2 + y^2 - 6x - 10y + 18 = 0 ) becomes ( (x - 3)^2 + (y - 5)^2 = 16 )
Finding Center and Radius from General Form
- After converting to standard form, (h, k) is the center and the radius is the square root of r^2.
- For ( (x - 3)^2 + (y - 5)^2 = 16 ), center is (3, 5), radius is 4.
Working with Coefficients
- If the general form has coefficients other than 1 for ( x^2 ) or ( y^2 ), divide the equation by that coefficient before completing the square.
Key Terms & Definitions
- Standard Form — ( (x - h)^2 + (y - k)^2 = r^2 )
- General Form — ( x^2 + y^2 + Dx + Ey + F = 0 )
- Center — The point (h, k) in standard form.
- Radius — The distance r from the center to the circle.
- Completing the Square — Method to rewrite quadratics as binomial squares.
Action Items / Next Steps
- Practice converting between general and standard forms using completing the square.
- Identify center and radius from standard form equations.
- Complete any assigned problems on circle equations.