Graph the following ellipse:
$4{(x-1)}^2+9{(y-2)}^2=36$

• A.
• B.
• C.
• D.

Answer: (D)

Verified step by step guidance

1. Start with the given ellipse equation: \(4\left(x-1\right)^2 + 9\left(y-2\right)^2 = 36\).
2. Divide both sides of the equation by 36 to write it in standard form: \(\frac{4\left(x-1\right)^2}{36} + \frac{9\left(y-2\right)^2}{36} = 1\), which simplifies to \(\frac{\left(x-1\right)^2}{9} + \frac{\left(y-2\right)^2}{4} = 1\).
3. Identify the center of the ellipse from the equation, which is at \((h, k) = (1, 2)\).
4. Determine the lengths of the semi-major and semi-minor axes: \(a^2 = 9\) so \(a = 3\), and \(b^2 = 4\) so \(b = 2\). Since \(a > b\), the major axis is horizontal.
5. Plot the ellipse centered at \((1, 2)\) with vertices at \((1 \pm 3, 2)\), which are \((4, 2)\) and \((-2, 2)\), and co-vertices at \((1, 2 \pm 2)\), which are \((1, 4)\) and \((1, 0)\). Then sketch the ellipse passing through these points.