https://file.aiquickdraw.com/mathbot/file/1725845716622rksq8q2w.jpg

The given image shows a determinant of a 2x2 matrix:

\[
\begin{vmatrix}
x - 2 & y + 3 \\
-y - 3 & x - 2
\end{vmatrix} = 0
\]

For a 2x2 matrix, the determinant is calculated as:

\[
\text{determinant} = (x - 2)(x - 2) - (y + 3)(-y + 3)
\]

Now, let's expand and simplify:

1. \((x - 2)(x - 2) = (x - 2)^2 = x^2 - 4x + 4\)
2. \((y + 3)(-y + 3) = - (y^2 - 9)\) which simplifies to \(-y^2 + 9\)

Now, the determinant equation becomes:

\[
x^2 - 4x + 4 + y^2 - 9 = 0
\]

Simplifying further:

\[
x^2 - 4x + y^2 - 5 = 0
\]

This is the equation of a conic, specifically a circle, that is centered at \((2, 0)\) with a radius of \(\sqrt{5}\).

### Final Equation:

\[
x^2 - 4x + y^2 = 5
\]

Would you like more details or have any further questions?

Here are 5 related questions you could explore:

1. What is the general formula for finding the determinant of a 3x3 matrix?
2. How does the sign of the determinant affect the properties of the matrix?
3. What is the geometric interpretation of the determinant in terms of linear transformations?
4. How does changing the coefficients of \(x\) and \(y\) affect the conic section's shape?
5. Can we classify this conic section further into ellipses, hyperbolas, or parabolas based on this equation?

**Tip:** When working with determinants, always pay attention to the sign of each term, especially when dealing with products of terms with signs.

Solve the determinant equation involving a 2x2 matrix as shown in the image: | x - 2     y + 3 | = 0 | -y + 3   x - 2 |.

Learn how to solve a determinant of a 2x2 matrix and reduce it to a circle equation. Explore the key steps and formulas used, including determinant properties and conic sections. Suitable for students in Grades 10-12.

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