Certainly! Let's go through the step-by-step solution to the given equation:
The given equation is:
[tex]\[
\frac{x^2}{16} + \frac{y^2}{12} = 1
\][/tex]
This is the standard form of an ellipse equation. To understand how this represents an ellipse and what its characteristics are, let's break it down:
### Step 1: Identify the Form of the Equation
The general form for the equation of an ellipse centered at the origin [tex]\((0,0)\)[/tex] is:
[tex]\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\][/tex]
### Step 2: Compare with the Standard Form
Compare the given equation with the standard form:
[tex]\[
\frac{x^2}{16} + \frac{y^2}{12} = 1
\][/tex]
From this comparison, we can deduce:
[tex]\[
\frac{x^2}{16} \implies a^2 = 16 \implies a = \sqrt{16} = 4
\][/tex]
[tex]\[
\frac{y^2}{12} \implies b^2 = 12 \implies b = \sqrt{12} = 2\sqrt{3}
\][/tex]
### Step 3: Interpret the Parameters
- [tex]\(a = 4\)[/tex]: This represents the semi-major axis of the ellipse.
- [tex]\(b = 2\sqrt{3}\)[/tex]: This represents the semi-minor axis of the ellipse.
### Step 4: Sketch and Analyze the Ellipse
Knowing the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we can describe the ellipse:
- The major axis is along the x-axis, because [tex]\(a > b\)[/tex].
- The minor axis is along the y-axis.
### Step 5: Write the Key Features
- Center: The ellipse is centered at [tex]\((0,0)\)[/tex].
- Vertices: The vertices of the ellipse are located at [tex]\((\pm a, 0)\)[/tex] and [tex]\((0, \pm b)\)[/tex].
- Vertices on the x-axis: [tex]\((\pm 4, 0)\)[/tex] or [tex]\((4, 0)\)[/tex] and [tex]\((-4, 0)\)[/tex]
- Vertices on the y-axis: [tex]\((0, \pm 2\sqrt{3})\)[/tex] or [tex]\((0, 2\sqrt{3})\)[/tex] and [tex]\((0, -2\sqrt{3})\)[/tex]
### Step 6: Concluding the Analysis
The equation [tex]\(\frac{x^2}{16} + \frac{y^2}{12} = 1\)[/tex] represents an ellipse centered at the origin with a semi-major axis of 4 units along the x-axis and a semi-minor axis of [tex]\(2\sqrt{3}\)[/tex] units along the y-axis. The vertices of this ellipse are located at the points [tex]\((4, 0)\)[/tex], [tex]\((-4, 0)\)[/tex], [tex]\((0, 2\sqrt{3})\)[/tex], and [tex]\((0, -2\sqrt{3})\)[/tex].
Therefore, the analysis and features from the given ellipse equation have been clearly established.
