Question

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

center at \\((0, 7)\\); \\(r = 9\\)

Explanation:

🆕 New Concept Discovered: Equation of a Circle
All points at a fixed distance from a center point.

Step 1: Identify the standard form equation

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

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

Step 2: Substitute the given values

We are given:

• Center \( (h, k) = (0, 7) \)
• Radius \( r = 9 \)

Substitute these values into the standard form equation:

$$ (x - 0)^2 + (y - 7)^2 = 9^2 $$

Step 3: Simplify the equation

Simplify the terms:

• \( (x - 0)^2 = x^2 \)
• \( 9^2 = 81 \)

This gives the simplified standard form:

$$ x^2 + (y - 7)^2 = 81 $$

Answer:

$$ x^2 + (y - 7)^2 = 81 $$