Problem 8 - 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 \(x^{2}+y^{2}+D x+E y+F=0\). $$ \text { Center at the origin and } r=1 \quad x^{2}+y^{2}-1=0 $$

Step-by-Step Solution

Answer

The equation is \(x^2 + y^2 - 1 = 0\).

Step 1: Understand the Problem

You are asked to write the equation of a circle given its center and radius. The center is at the origin (0,0) and the radius, denoted as \(r\), is 1.

Step 2: Recall the Standard Form of a Circle

The standard form of the equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius.

Step 3: Substitute Values into the Standard Form

Since the center is at \( (0,0) \) and \( r = 1 \), substitute these values into the standard form equation: \((x-0)^2 + (y-0)^2 = 1^2\). Simplify to get \(x^2 + y^2 = 1\).

Step 4: Convert to the Required Form

Rewrite \(x^2 + y^2 = 1\) in the requested form \(x^2 + y^2 + Dx + Ey + F = 0\). This gives \(x^2 + y^2 + 0x + 0y - 1 = 0\). Thus, \(D = 0\), \(E = 0\), \(F = -1\).

Step 5: Verification

Ensure the form matches the given condition by verifying that the circle's equation is correctly presented in the specified format.

Key Concepts

standard form of a circle

When working with circles in mathematics, the standard form of a circle's equation is pivotal. This is expressed as \((x-h)^2 + (y-k)^2 = r^2\). Here:

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

This form allows us to easily identify both the center and radius of the circle just by looking at the equation. To apply this, you can substitute the known values of the center and radius into this formula.
For instance, if we know the center is at a point \((h,k)\) and the radius is \( r \), we can put these values directly into the formula to get our circle's equation. It's simple but powerful!
Once you learn to recognize this form, it becomes an invaluable tool in solving and understanding geometric problems related to circles.

center at origin

Placing the center of a circle at the origin simplifies its equation significantly. The origin in coordinate geometry is the point \((0,0)\). If a circle's center is at the origin, it simplifies our standard equation to \( (x-0)^2 + (y-0)^2 = r^2 \).
This means our equation becomes \( x^2 + y^2 = r^2 \). There's no shifting, as the circle is perfectly centered on the point where the x-axis and y-axis intersect. This is perhaps the simplest form of a circle's equation, emphasizing symmetrical properties because everything is balanced around the origin itself.
Having the circle centered at the origin is not only straightforward but also highlights the neatness of constructing and analyzing geometric shapes from a central, critical point.

radius of a circle

The radius of a circle is a fundamental measurement of its size and shape. It is the distance from the center of the circle to any point on its boundary. In mathematical terms, the radius is represented by the letter \( r \).
When using the standard form of a circle, \((x-h)^2 + (y-k)^2 = r^2\), the square of the radius, \(r^2\), appears in the equation, guiding us to the circle's possible size. This allows us to calculate the area and circumference of the circle in further applications, using formulas like:

• Area: \( \pi r^2 \)
• Circumference: \( 2\pi r \)

Knowing the radius gives us not only an understanding of the circle's dimensions but also its position and influence within a coordinate plane or when solving practical problems involving circles.