Certainly! Let's solve the given equation step-by-step.
The given equation is:
[tex]\[
\frac{x^2}{5^2} + \frac{y^2}{3^2} = 1
\][/tex]
First, we simplify the denominators:
[tex]\[
5^2 = 25 \quad \text{and} \quad 3^2 = 9
\][/tex]
Inserting these into the equation, we get:
[tex]\[
\frac{x^2}{25} + \frac{y^2}{9} = 1
\][/tex]
Next, we rewrite this equation in a more standard form:
[tex]\[
\frac{x^2}{25} + \frac{y^2}{9} = 1
\][/tex]
Now, let us transform this equation to make it look like an equation used in identifying conic sections (like ellipses). Notice that an equation of this form:
[tex]\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\][/tex]
is the standard form of an ellipse, where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the semi-major and semi-minor axes.
Here, [tex]\( a^2 = 25 \)[/tex], so:
[tex]\[
a = \sqrt{25} = 5
\][/tex]
And [tex]\( b^2 = 9 \)[/tex], so:
[tex]\[
b = \sqrt{9} = 3
\][/tex]
Given these values, the equation:
[tex]\[
\frac{x^2}{25} + \frac{y^2}{9} = 1
\][/tex]
describes an ellipse centered at the origin (0,0) with a semi-major axis of length 5 and a semi-minor axis of length 3.
As a more concise representation, our final and simplified equation remains:
[tex]\[
\frac{x^2}{25} + \frac{y^2}{9} - 1 = 0
\][/tex]
This is the exact form representing an ellipse.
