Question

graph the circle.
\\x^2 + y^2 - 5 = 0\\

Explanation:

🆕 New Concept Discovered: Standard Equation of a Circle
Finding the center and radius to graph a circle

Step 1: Rewrite the equation in standard form

The standard form of a circle centered at the origin \( (0,0) \) with radius \( r \) is:

$$ x^2 + y^2 = r^2 $$

We start with the given equation:

$$ x^2 + y^2 - 5 = 0 $$

Add \( 5 \) to both sides to isolate the variable terms:

$$ x^2 + y^2 = 5 $$

Step 2: Identify the center and radius

Comparing \( x^2 + y^2 = 5 \) to the standard form \( x^2 + y^2 = r^2 \):

• Center: \( (0, 0) \)
• Radius squared: \( r^2 = 5 \)
• Radius: \( r = \sqrt{5} \approx 2.24 \)

Step 3: Match with the correct graph

• First Graph: The circle is centered at \( (0,0) \) and passes slightly beyond \( 2 \) units on the axes (at approximately \( 2.24 \)).
• Second Graph: The circle is centered at \( (0,0) \) but has a radius of \( 5 \) units (passing through \( (5,0) \), \( (0,5) \), etc.).

Since our radius is \( r = \sqrt{5} \approx 2.24 \), the first graph is the correct representation.

Answer:

The correct graph is the first graph (on the left), which shows a circle centered at the origin with a radius of approximately \( 2.24 \) (since \( r = \sqrt{5} \)).