%5E2%2B(y-2)%5E2%3D9-in-general-form.jpeg "(x-4)^2+(y-2)^2=9 In General Form")
From Standard to General Form: (x - 4)² + (y - 2)² = 9
The equation (x - 4)² + (y - 2)² = 9 represents a circle in standard form. To convert it to general form, we need to expand the squares and rearrange the terms.
Steps to Convert:
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Expand the squares: (x - 4)² + (y - 2)² = 9 x² - 8x + 16 + y² - 4y + 4 = 9
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Move the constant term to the right side: x² - 8x + y² - 4y = 9 - 16 - 4
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Simplify the right side: x² - 8x + y² - 4y = -11
The equation (x² - 8x + y² - 4y = -11) is now in general form.
What is General Form?
The general form of a circle's equation is:
x² + y² + Dx + Ey + F = 0
Where D, E, and F are constants.
Why is General Form Useful?
While standard form makes it easy to identify the center and radius of a circle, the general form is useful for:
- Finding the equation of a circle given its center and a point on the circle.
- Finding the equation of a circle given three points on the circle.
- Manipulating equations to determine other properties of the circle, such as its diameter or circumference.
In summary, understanding how to convert between standard and general form of a circle's equation allows for a more versatile and comprehensive understanding of its properties.