Problem 57 - Intermediate Algebra

Question

Find the equation of the circle that passes through the origin and has its center at \((-4,3)\).

Step-by-Step Solution

Answer

The equation of the circle is \((x+4)^2 + (y-3)^2 = 25\).

Step 1: Identify the General Equation of a Circle

The general equation of a circle with center \((h, k)\) and radius \(r\) is given by \((x-h)^2 + (y-k)^2 = r^2\). Here, the center \((h, k)\) is \((-4, 3)\), so the equation becomes \((x+4)^2 + (y-3)^2 = r^2\).

Step 2: Substitute the Given Point

Since the circle passes through the origin, or the point \((0, 0)\), we substitute these values into the circle's equation. This gives: \((0+4)^2 + (0-3)^2 = r^2\).

Step 3: Solve for the Radius \(r\)

Simplify the substitution result from Step 2, \(4^2 + 3^2 = r^2\), which simplifies to \(16 + 9 = r^2\). Therefore, \(r^2 = 25\).

Step 4: Write the Specific Equation of the Circle

Substitute \(r^2 = 25\) back into the circle’s equation from Step 1: \((x+4)^2 + (y-3)^2 = 25\). This is the equation of the circle that passes through the origin with center \((-4, 3)\).

Key Concepts

Center and Radius

When working with circles in geometry, two fundamental components to understand are the center and the radius. The **center** of a circle is the point from which all points on the circle are equidistant. It's like the heart of the circle. In our exercise, this center has the coordinates

• Center coordinates:
  • \(-4\) on the x-axis
  • \(3\) on the y-axis

The **radius** is the distance from this center to any point on the circle. Knowing the center and the circle passing through the origin, we can find the radius using the distance formula.
When a circle's equation and its center are known, it becomes straightforward to find the radius if it passes through any specific point.

General Equation of a Circle

The general equation of a circle provides a way to mathematically represent any circle on the coordinate plane. This equation is expressed as:
\[(x - h)^2 + (y - k)^2 = r^2\]

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

For our specific case, the center is

• \((-4, 3)\)

Thus, it leads to the equation:

• \((x + 4)^2 + (y - 3)^2 = r^2\)

This equation accounts for how the circle looks on a graph, and shows its size and position in relation to the coordinate axes. It becomes a powerful tool in understanding the circle's geometry.

Solving for Radius

To solve for the radius of the circle, it's important to use the specific point that lies on the circle. In this exercise, the point is the origin

• \((0, 0)\)

When the circle passes through this point, we substitute it into the circle's equation:
\[(0 + 4)^2 + (0 - 3)^2 = r^2\]This simplifies to:

• \[16 + 9 = r^2\]
• \[r^2 = 25\]

Finally, solving for \(r\) itself requires taking the square root of 25, which results in

• \(r = 5\)

This means the radius is 5 units long.
By understanding this process, you can easily find the radius when given the circle's center and a point on the circle.