Problem 74 - Algebra 2

Question

Write an equation of a circle with the given center and radius. center \((-4,7),\) radius 11

Step-by-Step Solution

Answer

The equation of the circle with the given center and radius is \((x+4)^2 + (y-7)^2 = 121\).

Step 1: Identify the Circle Parameters

Identify the parameters of the circle equation, which are the center (h,k) and radius r. Here, h=-4, k=7, and r=11.

Step 2: Substitute Parameters into Standard Circle Equation

Substitute the values of h, k, and r into the standard form of a circle equation, \((x-h)^2 + (y-k)^2 = r^2\). So the equation becomes \((x - (-4))^2 + (y-7)^2 = 11^2\).

Step 3: Simplify the Equation

Simplify the equation from step 2. The equation then becomes \((x+4)^2 + (y-7)^2 = 121\).

Key Concepts

Understanding the Standard Form of a Circle

The standard form of a circle's equation is pivotal for understanding how circles work in coordinate geometry. It's expressed as \((x-h)^2 + (y-k)^2 = r^2\). This format is built upon three key components:

• \(x\) and \(y\) are the coordinates of any point on the circle.
• \(h\) and \(k\) are the coordinates of the circle's center.
• \(r\) represents the circle’s radius.

The equation essentially conveys that every point \((x, y)\) on the circle is exactly \(r\) distance away from the center \((h, k)\).

This form acts as a foundation for analyzing circles, ensuring calculations with centers and varying radii can be done with ease. Understanding this form sets the stage for graphing and further algebraic manipulations.

Exploring the Center-Radius Form

The center-radius form is just another name for the standard form. It focuses on explicitly highlighting the circle’s features:

• Center: This is \((h, k)\).
• Radius: Denoted as \(r\).

When you have a circle centered at \(-4, 7\) with a radius of 11, just plug them into the equation form:

\[(x - (-4))^2 + (y - 7)^2 = 11^2\]

Which simplifies to:

\[(x + 4)^2 + (y - 7)^2 = 121\]

The center-radius format is user-friendly, offering a straightforward method to write and understand circle equations. It focuses on what makes the circle unique – its center and radius.

Simplifying the Circle Equation

Circle equation simplification is a key process in making equations more usable. It involves reducing the equation to its simplest form for clarity or further calculation. Starting from:

\[(x - (-4))^2 + (y - 7)^2 = 11^2\]

which becomes:

\[(x + 4)^2 + (y - 7)^2 = 121\]

, this step is essential, as it makes the circle's properties clearer and the equation easier to interpret.

This simplicity allows for quick graphing, easier calculation in analytics, and a more comprehensible format for solving geometry problems. The simplification doesn't change the circle's characteristics but enhances understanding and usability.