https://file.aiquickdraw.com/mathbot/file/1725483312435owr2dt8v.jpg

The equation provided in the image is:

\[
4x^2 + 16y^2 = 52
\]

To express this equation in the standard form of an ellipse, we should first divide both sides of the equation by 52 to get 1 on the right-hand side:

\[
\frac{4x^2}{52} + \frac{16y^2}{52} = \frac{52}{52}
\]

This simplifies to:

\[
\frac{x^2}{13} + \frac{y^2}{3.25} = 1
\]

Next, let's rewrite it using fractions with integers in the denominators:

\[
\frac{x^2}{\frac{52}{4}} + \frac{y^2}{\frac{52}{16}} = 1
\]

Thus, the standard form of the ellipse is:

\[
\frac{x^2}{13} + \frac{y^2}{\frac{13}{4}} = 1
\]

This equation represents an ellipse centered at the origin \((0, 0)\) with the semi-major axis along the \(x\)-axis and semi-minor axis along the \(y\)-axis.

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. How can you determine the lengths of the major and minor axes of this ellipse?
2. What is the difference between an ellipse and a circle in terms of their equations?
3. How do you find the foci of this ellipse?
4. What role does the discriminant play in identifying conic sections?
5. How do you graph an ellipse given its standard equation?

**Tip:** When converting an ellipse equation to standard form, always start by making the right-hand side equal to 1 for easier identification of axes and other properties.

Learn how to convert the ellipse equation 4x^2 + 16y^2 = 52 into its standard form. This detailed solution covers key concepts in analytic geometry, including the standard form of an ellipse and properties of conic sections. Suitable for students in Grades 10-12.

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