Solve the general form 25x² + 9y² = 225 to its ellipse standard form

The given equation is in the general form of a conic section:

\[
25x^2 + 9y^2 = 225
\]

To convert this into the standard form of an ellipse, we will divide the entire equation by 225 to normalize it:

\[
\frac{25x^2}{225} + \frac{9y^2}{225} = \frac{225}{225}
\]

Simplifying the fractions:

\[
\frac{x^2}{9} + \frac{y^2}{25} = 1
\]

This is now the standard form of an ellipse:

\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]

where \( a^2 = 9 \) and \( b^2 = 25 \), giving us \( a = 3 \) and \( b = 5 \).

Thus, the equation of the ellipse in standard form is:

\[
\frac{x^2}{9} + \frac{y^2}{25} = 1
\]

This represents an ellipse centered at the origin \((0,0)\), with the semi-major axis \( b = 5 \) along the y-axis, and the semi-minor axis \( a = 3 \) along the x-axis.

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Would you like more details on ellipses or have any questions?

Here are 5 related questions:
1. What is the general equation of an ellipse centered at the origin?
2. How do we determine the orientation of the major axis of an ellipse?
3. Can the semi-major and semi-minor axes of an ellipse be equal?
4. How can we graph an ellipse given its standard form equation?
5. What are the foci of an ellipse, and how are they calculated?

**Tip**: In an ellipse, the major axis corresponds to the largest denominator in the standard form equation.

Convert the equation 25x² + 9y² = 225 into its standard ellipse form. This step-by-step guide explores conic sections, the transformation process, and key concepts like the semi-major and semi-minor axes. Ideal for Grades 10-12 students.

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