Question

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

center at \\((3, 4)\\); \\(r = 2\\)

Explanation:

🆕 New Concept Discovered: Equation of a Circle
Writing circles in standard and general forms

Step 1: Identify standard form

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

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

Given:

• Center \( (h, k) = (3, 4) \)
• Radius \( r = 2 \)

Substitute these values into the standard form:

$$ (x - 3)^2 + (y - 4)^2 = 2^2 $$

$$ (x - 3)^2 + (y - 4)^2 = 4 $$

Step 2: Expand and simplify to general form

To write the equation in general form \( x^2 + y^2 + Dx + Ey + F = 0 \), expand the squared binomials:

$$ (x^2 - 6x + 9) + (y^2 - 8y + 16) = 4 $$

Combine like terms and move all terms to the left side:

$$ x^2 + y^2 - 6x - 8y + 25 = 4 $$

$$ x^2 + y^2 - 6x - 8y + 21 = 0 $$

Answer:

$$ x^2 + y^2 - 6x - 8y + 21 = 0 $$