Question

In Problems 39-44, find the standard form of the equation of each circle. Center at the origin and containing the point (-2,3)

Short answer

The standard form of the equation is \( x^2 + y^2 = 13 \).

Step-by-step solution

Understand the Standard Form of a Circle

The standard form of the equation of a circle is given by \( (x - h)^2 + (y - k)^2 = r^2 \). Here, \((h, k)\) is the center of the circle and \(r\) is the radius.

Identify the Center

The center of the circle is given in the problem as the origin, which means \((h, k) = (0,0)\). So, substituting \(h\) and \(k\) into the equation gives us \( x^2 + y^2 = r^2 \).

Use the Given Point to Find the Radius

The circle contains the point \((-2,3)\). Use this point to find the radius by substituting \(x = -2\) and \(y = 3\) into the simplified equation \( x^2 + y^2 = r^2 \).

Calculate the Radius

Substituting the coordinates of the point into the equation, \( (-2)^2 + 3^2 = r^2 \) which simplifies to \( 4 + 9 = r^2 \). So, \( 13 = r^2 \).

Write the Standard Form Equation

Now that the value of \(r^2\) is determined, substitute \(r^2 = 13\) back into the standard form equation to get \( x^2 + y^2 = 13 \).

Key concepts

Standard Form Equation of a Circle

The standard form of a circle is a key representation in coordinate geometry. Knowing this form allows us to easily understand and visualize circles on a coordinate plane. The standard form equation is given by:
\((x - h)^2 + (y - k)^2 = r^2\)
Here, \((h, k)\) represents the center of the circle, and \(r\) is the radius.
By understanding this form, solving problems involving circles becomes straightforward. In this equation:

• \(h\) and \(k\): coordinates of the circle's center
• \(r\): radius of the circle

This equation allows us to quickly determine the size and position of the circle on the coordinate plane. Using specific values for the center and radius, we can graph the circle precisely.

Finding the Radius

The radius is a crucial part of the circle's equation. It measures the distance from the center to any point on the circle. When a point on the circle and the center are provided, we can find the radius using the distance formula.
To find the radius of a circle centered at the origin containing the point \((-2,3)\):

• Substitute the coordinates of the given point into the simplified equation \(x^2 + y^2 = r^2\).
• Calculate the squared values: \((-2)^2 = 4\) and \(3^2 = 9\).
• Add these values: \(4 + 9 = 13\).
• Therefore, \(r^2 = 13\), and the radius \(r\) is \(\sqrt{13}\).

By following this method, we can find the radius if we know a point on the circle and its center coordinates.

Understanding the Center of a Circle

The center of a circle is a vital point in its equation. This point is denoted by \((h, k)\) in the standard form equation.
In our problem, the center is at the origin, meaning \((h, k) = (0, 0)\).
When the circle is centered at the origin, the equation simplifies to:

• \(x^2 + y^2 = r^2\)

This simplification makes it easier to work with the equation. The center of the circle defines its position on the coordinate plane. Every point on the circle is equidistant from this central point, defining the uniform shape of the circle.

Role of Coordinate Geometry

Coordinate geometry is essential for understanding and working with equations of circles. It uses algebraic equations to represent geometric shapes like circles on a coordinate plane.
Consider the standard form of the circle equation:

• \((x - h)^2 + (y - k)^2 = r^2\)

Coordinate geometry helps in:

• Plotting the circle accurately on a graph.
• Determining points on the circle using the equation.
• Understanding the relationship between algebraic equations and geometric figures.

For example, with a circle centered at the origin and containing the point \((-2, 3)\), coordinate geometry allows us to find the radius and visualize the circle accurately on the graph. Mastering these concepts enables solving complex geometry problems with ease.