Problem 5 - Algebra 2

Question

Write an equation of a circle with the given center and radius. Check your answers. $$ (1,-3), 10 $$

Step-by-Step Solution

Answer

The equation of the circle with center at (1,-3) and radius 10 is \((x-1)^2 + (y+3)^2 = 100\).

Step 1: Define center and radius

The given center of the circle is (1,-3) and the radius is 10.

Step 2: Substitute center and radius in the formula

Substitute the coordinates of the center and the radius in the circle equation formula \( (x-h)^2 + (y-k)^2 = r^2 \). So the required circle equation is \[ (x-1)^2 + (y+3)^2 = 10^2 \].

Step 3: Simplify

Simplifying the right side of the equation gives us the final form of the equation: \[ (x-1)^2 + (y+3)^2=100 \].

Key Concepts

Center and Radius

The center and radius of a circle are foundational elements that define its position and size. The center of a circle is a point that is equidistant from all points on the circle. It is often denoted as

• \( h \) for the x-coordinate
• \( k \) for the y-coordinate

If you think of the circle as the face of a clock, the center is the mechanical point from which both the minute and hour hands originate. Hence, in our problem, the center is given at the point \((1, -3)\).
The radius is the constant distance from the center to any point on the circle. A longer radius increases the circle's size, while a shorter radius decreases it. In this example, the radius is 10. Such a substantial radius indicates a relatively large circle. Remember, the radius must always be a positive value since it represents a distance.

Circle Equation Formula

The circle equation formula enables us to algebraically describe any circle when we know its center and radius. The standard form of a circle's equation is: \[(x-h)^2 + (y-k)^2 = r^2\]Here:

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

This formula is derived from the Pythagorean Theorem. It states that the sum of squares of a circle's radius must equal the squared sum of the differences in x's and y's from their respective center coordinates.
For our exercise, plugging in the values of the center \((h, k) = (1, -3)\) and radius \(r = 10\), we arrive at: \[(x-1)^2 + (y+3)^2 = 10^2\]
This initial substitution prepares our equation to accurately represent the circle.

Simplification of Equations

Simplifying an equation is an integral step to ensure clarity and ease of use. Simplification involves performing basic arithmetic or algebraic manipulations that make equations straightforward to work with. For circle equations, simplification primarily involves carrying out square root computations and reducing equations to cleaner forms.
In our case, the simplification targets the right side of the equation, where \(10^2\) is computed. Thus, simplifying \(10^2\) gives\(100\), leading to a cleaner and straightforward circle equation: \[(x-1)^2 + (y+3)^2 = 100\]
This step ensures that the equation is easily recognizable and checkable without complicated components. Always verify your simplification to ensure accuracy in representing your original terms.