Problem 57 - Contemporary Precalculus

Question

Find the equation of the circle with given center and radius \(r\). $$(0,0) ; \quad r=\sqrt{3}$$

Step-by-Step Solution

Answer

Answer: The equation of the circle is x² + y² = 3.

Step 1: Write down the circle equation formula

The circle equation formula is $$(x-a)^2 + (y-b)^2 = r^2$$

Step 2: Substitute the given center and radius

Use the given center (0,0) and radius \(\sqrt{3}\), and substitute their values into the circle equation formula: $$(x-0)^2 + (y-0)^2 = (\sqrt{3})^2$$

Step 3: Simplify the equation

Simplify the equation by removing unnecessary (0) value and squaring the radius: $$x^2 + y^2 = 3$$

Step 4: Final answer

The equation of the circle with center (0,0) and radius \(\sqrt{3}\) is: $$x^2 + y^2 = 3$$

Key Concepts

Center of a Circle

The center of a circle is the fixed point from which every point on the circumference is equidistant.
The center is a critical component when determining the equation of a circle.
In coordinate geometry, the center of a circle is often represented by the coordinates \(a, b\).

• In our example, the center is given as (0,0).
• This means the circle is centered at the origin of the coordinate plane.

Whenever you plug the center coordinates into the equation of a circle, it simplifies the process.
If the circle's center is at the origin, the equation becomes simpler because both \(a\) and \(b\) are zero.
This helps in reducing steps needed in calculation.

Radius

The radius of a circle is the distance from the center to any point on the circle.
It is a fundamental aspect as it dictates the size of the circle.
In the equation of a circle, this is represented by \(r\), which is the radius.

• For our circle, the radius is given as \(\sqrt{3}\).
• This tells us that every point on the circle is \(\sqrt{3}\) units away from the center.

Understanding the radius is crucial for fully grasping the circle's geometry.
When finding the equation, you'll square the radius, thereby introducing \(r^2\) into the equation.
For this problem, squaring the given radius \(\sqrt{3}\) results in 3, showing how the radius affects the equation.

Equation of a Circle

The equation of a circle mathematically represents all the points around a center at a specific radius.
It is a formula that integrates the center and radius into a standard structure:

• The general form is \( (x-a)^2 + (y-b)^2 = r^2 \).
• Where \(a\) and \(b\) are the center coordinates, and \(r\) is the radius.

In the given problem, the process involves plugging in our specific values.
Since our center is at (0,0) and radius is \(\sqrt{3}\):

• The standard form simplifies to \(x^2 + y^2 = (\sqrt{3})^2 \).
• Further simplifying gives \(x^2 + y^2 = 3\).

This is the complete equation for our circle, neatly representing its geometric properties.
Remember, this formula works universally for any circle when you know its center and radius.