To express the given equation \( X^2 + Y^2 = 5^2 \) in its general form, we need to first recognize that this is the equation of a circle. The equation of a circle in its standard form is:

\[
(X - h)^2 + (Y - k)^2 = r^2
\]

Where:
- \((h, k)\) is the center of the circle,
- \(r\) is the radius.

In your equation:
- The center \((h, k)\) is \((0, 0)\) (since there is no \((X-h)\) or \((Y-k)\) term).
- The radius \(r\) is 5.

The general form of the equation of a circle is obtained by expanding the standard form:

\[
X^2 + Y^2 + DX + EY + F = 0
\]

For a circle centered at the origin \((0, 0)\), \(D = 0\), \(E = 0\), and \(F = -r^2\). Therefore, for your equation:

\[
X^2 + Y^2 = 25
\]

We can rewrite it in general form as:

\[
X^2 + Y^2 + 0 \cdot X + 0 \cdot Y - 25 = 0
\]

Which simplifies to:

\[
X^2 + Y^2 - 25 = 0
\]

This is the general form of the equation for the given circle.

Would you like more details or have any questions?

Here are some related questions:
1. How would the general equation change if the circle were centered at \((2, -3)\)?
2. What is the general form of a circle with radius 7 centered at \((-4, 1)\)?
3. How can you convert a general equation of a circle back to its standard form?
4. How does the general equation of a circle differ from the general equation of an ellipse?
5. Can the general equation of a circle ever have non-zero coefficients for \(X\) or \(Y\) without being a circle?

**Tip:** When dealing with the general form of a conic section's equation, look out for the coefficients of \(X^2\) and \(Y^2\) to determine whether it represents a circle, ellipse, or another shape.

Explore how to derive the general equation of a circle from the standard form X^2 + Y^2 = 5^2. This detailed explanation covers the center, radius, and the conversion to general form. Suitable for high school students studying geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation