Problem 37 - College Algebra

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 circle equation with center at (-3,-1) and radius equal to \(\sqrt{3}\) is \((x + 3)^2 + (y + 1)^2 = 3\).

Step 1: Identify the values for h, k and r

From the given problem, the center of the circle (h,k) is (-3, -1) and radius (r) is \(\sqrt{3}\).

Step 2: Substitute the values into the standard circle's equation

Substitute h = -3, k = -1, and r = \(\sqrt{3}\) into the formula \((x - h)^2 + (y - k)^2 = r^2\). It becomes \((x - (-3))^2 + (y - (-1))^2 = (\sqrt{3})^2\).

Step 3: Simplify the equation

Simplify the equation to find the standard form of the circle equation. \((x + 3)^2 + (y + 1)^2 = 3\).

Key Concepts

Standard Form

The standard form of the equation of a circle is one of the easiest ways to express a circle mathematically in coordinate geometry. This form is given by the equation:\[(x - h)^2 + (y - k)^2 = r^2\]where

• \((h, k)\) represents the center of the circle in the coordinate plane
• \(r\) is the radius of the circle

Writing a circle's equation in this format helps neatly define its key properties at a glance: its center and its radius. This is why every time you have a center point and a radius, you can quickly substitute these values into the standard form equation to write the equation of the circle. For example, if you have a circle with center at \((-3, -1)\) and radius \(\sqrt{3}\), it neatly translates to:\[(x + 3)^2 + (y + 1)^2 = 3\] Thus allowing easy identification of its dimensions and position on the coordinate plane.

Center-Radius Form

The center-radius form is another term often used interchangeably with the standard form of a circle's equation. This concept highlights the focus on the circle's central point and size, simplifying the understanding of a circle's geometry. In the equation:\[(x - h)^2 + (y - k)^2 = r^2\]

• \((h, k)\) reveals the center of the circle
• \(r\), the radius, determines the extent or size of the circle

This form is particularly helpful when one needs to derive the circle's equation given the center and radius directly. For example, a circle centered at \((-3, -1)\) with a radius of \(\sqrt{3}\), is easily expressed by the center-radius formula:\[(x + 3)^2 + (y + 1)^2 = 3\] Using the center-radius form, identifying a circle's geometric characteristics becomes straightforward, thus making it a preferred choice among students and mathematicians.

Coordinate Geometry

Coordinate geometry is a branch of mathematics that deals with placing geometric figures in a coordinate plane and using algebraic equations to represent them. It provides a bridge connecting algebra and geometry and allows us to solve geometric problems numerically. In coordinate geometry, you can define shapes like circles, lines, and polygons using equations.
One of the shining examples in coordinate geometry is how we use the standard form of a circle's equation to describe its location and size in the coordinate plane. By setting a specific center and radius, you not only tell us where the circle is located but also provide vital information about its size.

• For instance, the equation \((x + 3)^2 + (y + 1)^2 = 3\) tells us that the circle is centered at \((-3, -1)\) with a radius of \(\sqrt{3}\)
• Coordinate geometry thus facilitates solving problems such as intersection, tangents, and positions of other shapes relative to each other using these equations.

Learning and understanding coordinate geometry is crucial in exploring how geometric shapes can interact and be analyzed in a coordinate space through equations.