Problem 36 - Precalculus Essentials

Question

Write the standard form of the equation of the circle with the given center and radius. Center \((-3,5), r=3\)

Step-by-Step Solution

Answer

The standard form of the equation of the circle with center (-3,5) and radius 3 is \( (x + 3)^2 + (y - 5)^2 = 9 \).

Step 1: Identifying the values

Identify the values given in the problem. Here, the center of the circle is given as (-3,5) so a=-3 and b=5. The radius is given as 3. Therefore, r=3.

Step 2: Substitute the values into the standard form

Substitute the identified values into the standard form of the circle's equation. So, replace a with -3, b with 5 and r with 3 in \( (x-a)^2 + (y - b)^2=r^2 \). This gives \( (x-(-3))^2 + (y - 5)^2=3^2 \).

Step 3: Simplify the equation

Simplify the equation obtained in the previous step. The expression 'x - -3' simplifies to 'x + 3'. The right hand side 3^2 equals 9. So, our final equation is \( (x + 3)^2 + (y - 5)^2 = 9 \).

Key Concepts

Standard Form of a Circle's Equation

The standard form of a circle's equation is a specific way of expressing the equation that highlights the circle’s center and radius.
This form looks like this: \((x-a)^2 + (y-b)^2 = r^2\). Here, \((a, b)\) represents the circle's center, and \(r\) is the radius.
The benefit of using standard form is its simplicity and clarity.

• It directly tells you the circle's center and radius.
• Easy to graph and visualize a circle.

For instance, if you have a circle centered at \((-3, 5)\) with a radius of \(3\), you substitute these values into the standard form.
This yields \((x + 3)^2 + (y - 5)^2 = 9\).
This equation makes it easy to spot the characteristics of the circle.

Center of a Circle

The center of a circle is a crucial point that determines its position on the coordinate plane.
The center is denoted by \((a, b)\) in the standard form of the circle's equation \((x-a)^2 + (y-b)^2 = r^2\).
This point doesn't alter the size of the circle, only its position.

• The circle's equation is symmetrical around its center.
• The distance from the center to any point on the circle is the same and is known as the radius.

In the given exercise, the center is \((-3, 5)\).
This means the circle is shifted left by 3 units and up by 5 units from the origin.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle.
In the standard form of a circle's equation \((x-a)^2 + (y-b)^2 = r^2\), \(r\) is the radius.
This value determines the size of the circle, or how wide it is.

• All points on the circle are exactly \(r\) units from the center.
• The radius is always a positive number.

In the exercise, the radius given is \(3\), which appears as \(r^2\) in the equation.
So, the equation \((x + 3)^2 + (y - 5)^2 = 9\) confirms that the radius squared is 9, leading to \(r = 3\).