Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$-x^{2}=y^{2}-25$$

Step-by-Step Solution

Answer

The equation represents a circle with center at (0,0) and radius 5.

Step 1: Rearrange the Equation

The given equation is \(-x^{2} = y^{2} - 25\). Start by rearranging the terms to group like variables together. After adding \(x^{2}\) to both sides, the equation becomes \(x^{2} + y^{2} = 25\).

Step 2: Identify the Standard Form

The rearranged equation \(x^{2} + y^{2} = 25\) matches the standard form of a circle equation: \((x-h)^{2} + (y-k)^{2} = r^{2}\), where \((h, k)\) is the center and \(r\) is the radius.

Step 3: Determine the Key Features

In the standard form \(x^{2} + y^{2} = 25\), the center of the circle is at \((0,0)\) and the radius is \(5\), since \(r^{2} = 25\) gives us \(r = 5\).

Step 4: Conclude the Graph Type

Since the equation \(x^{2} + y^{2} = 25\) fits the standard form of a circle and describes a set of points a constant distance from the center, we conclude this equation represents a circle.

Key Concepts

Equation of a Circle

In math, the equation of a circle is a special type of equation that gives us lots of information about a circle on a graph. Circles in math are defined as a set of all points that are a fixed distance, called the radius, from a fixed point, called the center.

The specific equation for a circle in the coordinate plane is:

• rac{(x-h)^2 + (y-k)^2 = r^2.

Here,

• (x, y) represents any point on the circle.
• (h, k) represents the center of the circle.
• and \(r\) is the radius.

Using the circle equation, we can find out many useful details about its size and location just from looking at these values. Isn't that neat?

Standard Form

The standard form of the equation of a circle is a helpful and concise way of writing it down. Remember our formula:

• rac{(x-h)^2 + (y-k)^2 = r^2.

The standard form makes it super easy to spot where the circle is on a graph and how big it is because:

• (h, k) tells us the center.
• The radius \(r\) tells us how far the circle extends from this center point.

When we face a different equation like \(-x^2 = y^2 - 25\), realizing this can't yet tell us about a circle, we rearrange it into the familiar form. As in our step-by-step solution, adding \(x^2\) brings us to \(x^2 + y^2 = 25\), which is clearly in a circle's standard form. This reorganization is key to connecting it to the circle in the graph.

Graph Identification

Understanding graph shapes begins when you take an equation and match it to known types, like circles. For example, from the equation \(x^2 + y^2 = 25\), we can quickly identify it's a circle, during the problem-solving step. This is because it aligns perfectly with the circle's standard form: \((x-h)^2 + (y-k)^2 = r^2\).

To identify:

• Look at the power of variables. When both \(x\) and \(y\) are squared and summed, think circle.
• Check for the presence of terms like \((x-h)^2\). If present, it matches the standard form for circles.
• Distinguish from parabolas, ellipses, and hyperbolas, which have different arrangements.

This way, knowing the standard form helps you correctly identify the graph as a circle, not a different conic shape.

Radius Computation

One of the simplest yet powerful details you can find from a circle's equation is the radius. In the standard equation \((x-h)^2 + (y-k)^2 = r^2\), converting to this form shows us the square of the radius. But how do we get \(r\)?
Just solve:

• Look at the constant on the right-hand side of the equation. Here it's 25, expressed as \(r^2\).
• To find \(r\) take the square root of 25.
  This provides \(r = 5\).

Remember, the computed radius tells us about the ring size leaping out from the circle's center, and is critical for drawing its boundary on the graph. As we found earlier, converting \(x^2 + y^2 = 25\) shows not just a circle location, but how far it runs around its midpoint.