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Equation of a Circle

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Equation of a Circle

Study Guide

Key Definition

A circle is defined as the set of all points that are equidistant from a central point in a plane. The equation is $(x-h)^2 + (y-k)^2 = r^2$

Important Notes

The center of the circle is $(h, k)$
The radius is denoted by $r$
The diameter is twice the radius: $2r$
The standard form of a circle's equation is $(x-h)^2 + (y-k)^2 = r^2$
If the center is at the origin, the equation simplifies to $x^2 + y^2 = r^2$

Mathematical Notation

$(x-h)^2 + (y-k)^2 = r^2$: standard form of a circle

$Δx = x - h$: horizontal displacement from center

$Δy = y - k$: vertical displacement from center

$√(Δx^2 + Δy^2) = r$: distance from (x,y) to center equals radius

Use Δx and Δy when applying the distance formula to derive the circle's equation.

Why It Works

The equation $(x-h)^2 + (y-k)^2 = r^2$ represents all points $(x, y)$ that are a distance $r$ from the center $(h, k)$.

Remember

The radius connects the center to any point on the circle: $(x-h)^2 + (y-k)^2 = r^2$

Quick Reference

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

Understanding Equation of a Circle

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Video explanation of this concept

Beginner

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Beginner Explanation

The equation $(x-h)^2 + (y-k)^2 = r^2$ comes from the distance formula. Any point $(x,y)$ on the circle is exactly $r$ units from the center $(h,k)$. Writing $√((x-h)^2 + (y-k)^2) = r$ and squaring gives $(x-h)^2 + (y-k)^2 = r^2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the equation of a circle with center at $(0, 0)$ and radius $5$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine a circular garden with a radius of $7$ meters. What is the equation of the boundary of the garden if the center is at $(2, 3)$?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Find the radius of a circle whose equation is $(x+1)^2 + (y-1)^2 = 36$.

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following represents a circle with center at $(-3, 4)$ and radius $6$?

Recap

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Review key concepts and takeaways