Problem 7 - Algebra for College Students

Question

For Problems 1-14, write the equation of each of the circles that satisfies the stated conditions. In some cases there may be more than one circle that satisfies the conditions. Express the final equations in the form \(x^{2}+y^{2}+D x+E y+F=0\). Center at the origin and \(r=7\) $$ x^{2}+y^{2}-49=0 $$

Step-by-Step Solution

Answer

The equation of the circle is \(x^2 + y^2 - 49 = 0\).

Step 1: Identify Circle Properties

The problem states that the circle is centered at the origin \(0, 0\) and has a radius \(r = 7\). A circle equation with these properties follows a standard form.

Step 2: Recall Circle Equation Format

The standard equation for a circle centered at \( (h, k) \) with a radius \( r \) is \( (x - h)^2 + (y - k)^2 = r^2 \). Since the center is at the origin \( (0, 0) \), this equation simplifies to \( x^2 + y^2 = r^2 \).

Step 3: Substitute Known Values

Insert the given radius \( r = 7 \) into the simplified equation: \( x^2 + y^2 = 7^2 \). This becomes \( x^2 + y^2 = 49 \).

Step 4: Rearrange to Required Form

The question asks for the equation in the form \(x^2 + y^2 + Dx + Ey + F = 0\). To match this form, set \(D = 0, E = 0, \) and move 49 to the other side of the equation. This results in \(x^2 + y^2 - 49 = 0\).

Key Concepts

Understanding the Standard Form Circle Equation

The standard form of a circle's equation is crucial for quickly understanding a circle's properties based on its mathematical expression. The equation is expressed as \[(x - h)^2 + (y - k)^2 = r^2\] where

• \((h, k)\) represents the center of the circle.
• \(r\) is the radius of the circle.

If a circle is centered at the origin, the equation dramatically simplifies because the origin is \((0,0)\). Thus, all terms involving \(h\) and \(k\) disappear, leading to the simpler form \(x^2 + y^2 = r^2\).This standard form is essential to easily identify fundamental features of a circle such as its center and radius, which are key to understanding its position and size in a coordinate plane.

Identifying the Radius of a Circle

The radius of a circle is a constant distance from the center of the circle to any point on its circumference. In the standard form circle equation, the radius is represented as \(r\) in \((x - h)^2 + (y - k)^2 = r^2\). To find the radius, you can rewrite the circle's equation in this standard form and identify \(r^2\); the square root of this value gives you the radius.For example, if you have \((x - 0)^2 + (y - 0)^2 = 49\), then \(r^2 = 49\), and solving for \(r\), you find \(r = 7\).Understanding the radius helps in visualizing how expansive the circle is and determining whether different points lie inside, outside, or on the perimeter of the circle.

Finding the Center of a Circle

The center of a circle is a point from which every part of the circle is equidistant. Recognizing the center helps in defining the circle's position within the two-dimensional plane. In a standard form circle equation like \((x - h)^2 + (y - k)^2 = r^2\), the values of \(h\) and \(k\) indicate the center.If a circle's equation is already simplified, as in \(x^2 + y^2 = 49\), it implies a center located at the origin \((0, 0)\). This is because the equation corresponds to \((x - 0)^2 + (y - 0)^2 = 49\). Knowing the center is vital for graphing the circle accurately and understanding how it relates to other geometric figures on the same coordinate plane.