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(x-3)^2+(y-4)^2=25 In General Form

2 min read Jun 17, 2024
(x-3)^2+(y-4)^2=25 In General Form

Converting the Equation (x-3)^2 + (y-4)^2 = 25 to General Form

The equation (x-3)^2 + (y-4)^2 = 25 represents a circle in standard form. The standard form of a circle is:

(x - h)^2 + (y - k)^2 = r^2

Where:

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

In our given equation, (h, k) = (3, 4) and r = 5. To convert it to general form, we need to expand the squares and rearrange the terms.

Steps to convert to general form:

  1. Expand the squares: (x - 3)^2 + (y - 4)^2 = 25 x^2 - 6x + 9 + y^2 - 8y + 16 = 25

  2. Move the constant term to the right side: x^2 - 6x + y^2 - 8y = 25 - 9 - 16 x^2 - 6x + y^2 - 8y = 0

  3. The equation in general form: x^2 + y^2 - 6x - 8y = 0

This is the general form of the equation of the circle. The general form is characterized by having all the terms on one side of the equation, with the constant term set to zero.

Key takeaway: The general form of the equation of a circle is Ax^2 + Ay^2 + Bx + Cy + D = 0, where A, B, C, and D are constants. In our example, A = 1, B = -6, C = -8, and D = 0.

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