Write the standard form equation of each ellipse centered at the origin.

• A. $\frac{x^2}{4}+y^2=1$
• B. $x^2+\frac{y^2}{16}=1$
• C. $x^2+y^2=16$
• D. $\frac{x^2}{16}+y^2=1$

Answer: (D)

Verified step by step guidance

1. Step 1: Identify the center of the ellipse. Since the ellipse is centered at the origin, the center is at (0, 0).
2. Step 2: Determine the lengths of the major and minor axes by looking at the intercepts on the x-axis and y-axis. The ellipse extends from -4 to 4 on the x-axis, so the length of the major axis is 8, and from -1 to 1 on the y-axis, so the length of the minor axis is 2.
3. Step 3: Calculate the values of \(a^2\) and \(b^2\), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. Since the major axis is horizontal, \(a = 4\) and \(b = 1\), so \(a^2 = 16\) and \(b^2 = 1\).
4. Step 4: Write the standard form equation of the ellipse centered at the origin with a horizontal major axis: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]. Substitute the values of \(a^2\) and \(b^2\) to get \[ \frac{x^2}{16} + \frac{y^2}{1} = 1 \].
5. Step 5: Simplify the equation if needed. Since \(\frac{y^2}{1} = y^2\), the final standard form equation is \[ \frac{x^2}{16} + y^2 = 1 \].