Math Problem Statement

Write the standard form of the equation and the general form of the equation of the circle with the radius of 3 and the center (-3,0).

Solution

The equation of a circle is typically represented in two forms: standard form and general form.

1. Standard Form of the Equation of a Circle

The standard form is: (xh)2+(yk)2=r2(x - h)^2 + (y - k)^2 = r^2 where:

  • (h,k)(h, k) is the center of the circle,
  • rr is the radius.

For this problem:

  • The center is (3,0)(-3, 0),
  • The radius is 33.

Substituting these values into the standard form: (x(3))2+(y0)2=32(x - (-3))^2 + (y - 0)^2 = 3^2 Simplifying: (x+3)2+y2=9(x + 3)^2 + y^2 = 9 So, the standard form of the equation is: (x+3)2+y2=9(x + 3)^2 + y^2 = 9

2. General Form of the Equation of a Circle

The general form is obtained by expanding the standard form and arranging it into: Ax2+Ay2+Bx+Cy+D=0Ax^2 + Ay^2 + Bx + Cy + D = 0

Starting with the standard form (x+3)2+y2=9(x + 3)^2 + y^2 = 9, expand it: (x2+6x+9)+y2=9(x^2 + 6x + 9) + y^2 = 9 Now subtract 9 from both sides: x2+6x+y2=0x^2 + 6x + y^2 = 0 So, the general form of the equation is: x2+y2+6x=0x^2 + y^2 + 6x = 0

Summary:

  • Standard Form: (x+3)2+y2=9(x + 3)^2 + y^2 = 9
  • General Form: x2+y2+6x=0x^2 + y^2 + 6x = 0

Would you like more details or have any questions?

Here are 5 related questions to explore:

  1. How do you find the center and radius from a circle's equation in general form?
  2. What is the equation of a circle with a radius of 5 and center at (2, -3)?
  3. How would the equation change if the radius doubled?
  4. How do you convert the general form of a circle to its standard form?
  5. How can you graph a circle from its standard form equation?

Tip: When converting between the standard and general forms of a circle, always complete the square if needed!

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Math Problem Analysis

Mathematical Concepts

Geometry
Equations of Circles

Formulas

Standard form of a circle: (x - h)^2 + (y - k)^2 = r^2
General form of a circle: Ax^2 + Ay^2 + Bx + Cy + D = 0

Theorems

Equation of a Circle

Suitable Grade Level

Grades 9-11