Math Problem Statement

If the circle has the center of -2 and +2 and as the radius of 3 what is the standard form and the general form of it

Solution

Solution

Standard Form of the Circle's Equation:

The equation of a circle in 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.

Given:

  • Center (2,2)(-2, 2): h=2h = -2, k=2k = 2.
  • Radius r=3r = 3.

Substitute the values:

(x(2))2+(y2)2=32(x - (-2))^2 + (y - 2)^2 = 3^2

(x+2)2+(y2)2=9(x + 2)^2 + (y - 2)^2 = 9

Thus, the standard form is:

(x+2)2+(y2)2=9(x + 2)^2 + (y - 2)^2 = 9


General Form of the Circle's Equation:

To convert the standard form to the general form, expand and simplify:

(x+2)2+(y2)2=9(x + 2)^2 + (y - 2)^2 = 9

Expand each term:

x2+4x+4+y24y+4=9x^2 + 4x + 4 + y^2 - 4y + 4 = 9

Combine like terms:

x2+y2+4x4y+8=9x^2 + y^2 + 4x - 4y + 8 = 9

Simplify:

x2+y2+4x4y1=0x^2 + y^2 + 4x - 4y - 1 = 0

Thus, the general form is:

x2+y2+4x4y1=0x^2 + y^2 + 4x - 4y - 1 = 0


Final Answer:

  • Standard Form: (x+2)2+(y2)2=9(x + 2)^2 + (y - 2)^2 = 9
  • General Form: x2+y2+4x4y1=0x^2 + y^2 + 4x - 4y - 1 = 0

Would you like further clarification or visualization of this solution?

Related Questions:

  1. What are the properties of a circle represented in general form?
  2. How do you find the radius and center from the general form of a circle?
  3. What are the steps to graph a circle using the standard form equation?
  4. How do transformations of a circle affect its standard form equation?
  5. Can a circle have a radius of zero? If so, what would its equation look like?

Tip:

Always verify the center and radius when switching between standard and general forms to avoid calculation errors.

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Math Problem Analysis

Mathematical Concepts

Geometry
Equations of a Circle

Formulas

Standard form of a circle: (x - h)^2 + (y - k)^2 = r^2
General form of a circle: x^2 + y^2 + Dx + Ey + F = 0

Theorems

Definition of a Circle

Suitable Grade Level

Grades 9-12