Problem 75 - College Algebra with Modeling and Visualization

Question

Find the standard equation of a circle that satisfies the conditions. Center \((0,0)\) with the point \((-3,-1)\) on the circle

Step-by-Step Solution

Answer

The equation is \(x^2 + y^2 = 10\).

Step 1: Understanding the Problem

We're asked to find the standard equation of a circle with its center at the origin \((0,0)\) and passing through the point \((-3,-1)\).

Step 2: Identifying the Standard Equation of a Circle

The standard equation of a circle with center \((h,k)\) and radius \(r\) is \((x-h)^2 + (y-k)^2 = r^2\). For this problem, the center \((h,k)\) is \((0,0)\).

Step 3: Substitute the Center into the Equation

With the center at \((0,0)\), the equation simplifies to \(x^2 + y^2 = r^2\).

Step 4: Calculate the Radius

The radius \(r\) is the distance from the center of the circle \((0,0)\) to a point on the circle, in this case \((-3,-1)\). The distance formula is given by \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 5: Apply the Distance Formula

Using the coordinates of the center \((0,0)\) and the point \((-3,-1)\), calculate the radius: \(r = \sqrt{(-3-0)^2 + (-1-0)^2} = \sqrt{9 + 1} = \sqrt{10}\).

Step 6: Complete the Equation with Calculated Radius

Plugging the radius back into the circle equation, we have \(x^2 + y^2 = 10\). This is the standard equation of the circle.

Key Concepts

Standard Equation of a Circle

The standard equation of a circle is a formula that represents the set of all points that are equidistant from a fixed center point. This distance is known as the radius. If you imagine the circle drawn in a coordinate plane, it is the circle's edge that holds all these equidistant points.

The generic form of this equation is \[(x-h)^2 + (y-k)^2 = r^2\]Here:

• \((h, k)\) is the center of the circle
• \(r\) is the radius

For a circle centered at the origin \((0, 0)\), the equation simplifies to \[x^2 + y^2 = r^2\]This means that all you need is the radius to find the equation of a circle when its center is at the origin.
Understanding this basic form helps when working through various circle-related problems.

Distance Formula

The distance formula is a powerful tool in coordinate geometry that allows us to measure the straight-line distance between two points in a plane. It is derived from the Pythagorean theorem and provides a way to calculate lengths in any coordinate plane.

The distance formula is given by:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Here:

• \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of two points

In terms of finding a circle's radius, this formula is essential. If you have a center point of a circle and any other point on the circle, the distance formula helps calculate the radius by measuring this exact distance.
Using the distance formula can be thought of as creating a right triangle between the two points, where the line segment connecting them is the hypotenuse.

Radius Calculation

The calculation of the radius is a fundamental part of writing the standard equation of a circle. The radius determines how far the circle extends from its center and thus is crucial for its representation.

To find the radius when given a center and a point on the circle, apply the distance formula:\[r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]With our exercise, the center is \((0, 0)\) and the point \((-3, -1)\) lies on the circle:

• Calculate: \(r = \sqrt{(-3 - 0)^2 + (-1 - 0)^2} = \sqrt{9 + 1} = \sqrt{10}\)

Once you know the radius, you can finish the circle's equation.
The ability to calculate the radius confidently allows for further exploration into more complex problems involving circles.