Problem 33 - Algebra 2

Question

Write an equation for the circle that satisfies each set of conditions. center \((-\sqrt{13}, 42),\) passes through the origin

Step-by-Step Solution

Answer

The equation is \((x + \sqrt{13})^2 + (y - 42)^2 = 1777\).

Step 1: Understanding the Circle Equation

A circle's equation in the standard form is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle, and \(r\) is the radius.

Step 2: Plug Center into the Equation

Given the center is \((-\sqrt{13}, 42)\), substitute \(h = -\sqrt{13}\) and \(k = 42\) into the equation. This gives us \((x + \sqrt{13})^2 + (y - 42)^2 = r^2\).

Step 3: Use the Point on the Circle

The circle passes through the origin, meaning \((0, 0)\) is a point on the circle. Substitute \(x = 0\) and \(y = 0\) into the equation \((x + \sqrt{13})^2 + (y - 42)^2 = r^2\) to find \(r^2\).

Step 4: Calculate the Radius Squared

Substitute into the equation: \((0 + \sqrt{13})^2 + (0 - 42)^2 = r^2\). This simplifies to \(13 + 1764 = r^2\), and therefore \(r^2 = 1777\).

Step 5: Write the Final Equation

Plug \(r^2 = 1777\) back into the circle equation: \((x + \sqrt{13})^2 + (y - 42)^2 = 1777\). This is the final equation of the circle.

Key Concepts

Standard Form of a Circle

To understand circles in mathematics, we often start with their equation in what is called the "standard form". This form is represented as \((x-h)^2 + (y-k)^2 = r^2\). Here, \((h, k)\) is the circle's center, and \(r\) stands for the radius.

There are many handy features in this form! It's easy to spot both the center and radius from the equation, making it straightforward to visualize the circle on a graph.

• \(h\) and \(k\): These are the coordinates of the circle's center.
• \(r^2\): This is the square of the radius, so \(r = \sqrt{r^2}\).

Knowing this pattern helps us quickly identify a circle's main components when given an equation. It also aids in the process of writing a circle equation if you know the circle's center and any point it passes through.

Radius Calculation

Calculating the radius of a circle is crucial for understanding its size.
Knowing the center and a point that the circle passes through, like the origin \((0, 0)\) in this case, allows us to determine the radius using the equation of the circle.

To find the radius for a circle passing through a particular point, follow these steps:

• Substitute the point's \(x\) and \(y\) values into the circle's equation.
• This gives \((0 + \sqrt{13})^2 + (0 - 42)^2 = r^2\).
• Simplify to obtain \(13 + 1764 = r^2\), which is \(r^2 = 1777\).

Hence, solving for \(r^2\) lets you find \(r\) by taking the square root. Here, the value comes to \(\sqrt{1777}\), though we often keep it as \(r^2\) in the equation for simplicity and precision.

Center of a Circle

The center of a circle is like its heart, determining where the circle is located on a grid.

In every circle equation, the terms \((h, k)\) in the standard form \((x-h)^2 + (y-k)^2 = r^2\) denote the center's coordinates.

For example, if the circle's center is given as \((-\sqrt{13}, 42)\), you can immediately substitute these values in place of \(h\) and \(k\):

• The equation becomes \((x + \sqrt{13})^2 + (y - 42)^2 = r^2\).
• The '+\sqrt{13}' indicates that the center is reaching leftwards from the y-axis, as it originated from \(-h\).
• The '-42' shifts it upwards in the plane.

By understanding this, you can visualize the circle's place just by looking at its equation, confirming both its center and symmetry.