Problem 37 - College Algebra Essentials

Question

write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(-3,-1), r=\sqrt{3} $$

Step-by-Step Solution

Answer

The standard form of the equation of the circle with center (-3,-1) and radius \( \sqrt{3} \) is \( (x+3)^2 + (y+1)^2 = 3 \).

Step 1: Identify and Substitute the Given Values

The circle's center is given as (-3,-1) and radius as \( \sqrt{3} \). These correspond to the values of h, k and r in the standard formula of the circle, respectively. We put these values in the formula \( (x-h)^2 + (y-k)^2 = r^2 \).

Step 2: Squaring the Radius

Next, square the radius. In this particular case, the radius is \( \sqrt{3} \). The square of \( \sqrt{3} \) is simply 3. Hence, our equation becomes \( (x+3)^2 + (y+1)^2 = 3 \).

Step 3: Final Equation

As a result, the equation of the circle in standard form with a center of (-3,-1) and a radius of \( \sqrt{3} \) is \( (x+3)^2 + (y+1)^2 = 3 \).

Key Concepts

Understanding the Standard Form Equation of a Circle

The standard form of a circle's equation is an essential concept in geometry. It provides a simple way to represent a circle using its center and radius, making it easier to visualize and analyze. The formula is expressed as: \[ (x-h)^2 + (y-k)^2 = r^2 \] Here,

• \( (x-h) \) and \( (y-k) \) represent horizontal and vertical shifts from the circle's center \((h, k)\), respectively.
• \( r \) is the radius of the circle.

This format shows how far every point \((x, y)\) on the circle is from its center, ensuring the distance (radius) remains consistent. This consistency forms the circular shape. Understanding this formula allows us to easily pinpoint the essential characteristics of any circle by simply identifying its center and radius.

Determining the Radius of a Circle

The radius is a fundamental measurement in a circle, indicating the distance from the center to any point on the circle. In the standard form equation, the radius squared \((r^2)\) is on the right-hand side. In mathematical terms, if the radius is given as \( \sqrt{3} \), then squaring it results in 3, which forms part of the equation. It's important to note:

• The radius is always non-negative, as it represents a distance.
• Squaring a square root removes the root, simplifying calculations in the equation of a circle.

If you keep these in mind, it becomes easier to work through problems involving circles, simplifying the calculations involved.

Identifying the Center of a Circle

The center of a circle is a key element in understanding its placement on the coordinate plane. In the standard form of a circle's equation, the center is represented by the coordinates \((h, k)\). For an equation given by \((x+3)^2 + (y+1)^2 = 3\), the center values are effectively hidden by minus signs in the form \((x-h)\) and \((y-k)\). Here:

• \(x+3 \) corresponds to \(x-h\), indicating \( h = -3 \).
• \(y+1 \) matches \(y-k\), meaning \( k = -1 \).

Thus, the center is located at \((-3, -1)\). Understanding these transformations helps you accurately plot the circle's location on a graph and is essential for solving more complex geometric problems. Remember, the center coordinates will invert the signs present in the equation.