Question

write an equation in general form of the circle with the given properties.

center at the origin; \\(r = 5\\)

Explanation:

🆕 New Concept Discovered: Equation of a Circle
Finding standard and general forms of a circle

Step 1: Identify the standard form equation

The standard form of the equation of a circle with center \( (h, k) \) and radius \( r \) is:

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

Given:

• Center at the origin: \( (h, k) = (0, 0) \)
• Radius: \( r = 5 \)

Substitute these values into the standard form:

$$ (x - 0)^2 + (y - 0)^2 = 5^2 $$

$$ x^2 + y^2 = 25 $$

Step 2: Convert to general form

The general form of the equation of a circle is written with all terms on one side, set equal to zero:

$$ x^2 + y^2 + Dx + Ey + F = 0 $$

To convert our equation to this form, subtract \( 25 \) from both sides:

$$ x^2 + y^2 - 25 = 0 $$

Answer:

$$ x^2 + y^2 - 25 = 0 $$