Certainly! Let's carefully analyze and solve the equation \(\frac{(x - c_1)^2}{2} + \frac{(y - c_2)^2}{2} = 1\).
Step-by-Step Solution:
1. Understanding the Ellipse Equation:
The given equation is of the form \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), which represents the equation of an ellipse. Here \(a\) and \(b\) are the semi-major and semi-minor axes, respectively, and \((h, k)\) is the center of the ellipse.
2. Rewrite the Given Equation:
The given equation looks like:
[tex]\[
\frac{(x - c_1)^2}{2} + \frac{(y - c_2)^2}{2} = 1
\][/tex]
3. Substitute \(c_1\) and \(c_2\):
Notice that the given equation seems to have missing center coordinates \(c_1\) and \(c_2\). Suppose that the center is at the origin (i.e., \(c_1 = 0\) and \(c_2 = 0\)), giving us:
[tex]\[
\frac{x^2}{2} + \frac{y^2}{2} = 1
\][/tex]
4. Analyze the Simplified Form:
This is already in the form of an ellipse centered at the origin with semi-major and semi-minor axes both having length \(\sqrt{2}\):
[tex]\[
\frac{x^2}{2} + \frac{y^2}{2} = 1
\][/tex]
5. Conclusion:
The equation \(\frac{x^2}{2} + \frac{y^2}{2} = 1\) correctly represents the ellipse centered at the origin where both \(a^2 = 2\) and \(b^2 = 2\).
Thus, the final step yields the result:
[tex]\[
\frac{x^2}{2} + \frac{y^2}{2} - 1 = 0
\][/tex]
Hence, this is the ellipse's equation simplified:
[tex]\[
x^2/2 + y^2/2 - 1 = 0
\][/tex]
