https://r2.erweima.ai/mathbot/image/1725458936239f4t26uvw.jpg

The equation given in the image is:

\[
y(1 - x^2)^{1/2} = x^2 + C
\]

To solve this equation, we can approach it by first isolating \( y \). Here's the step-by-step solution:

1. **Isolate \( y \)**:
   \[
   y = \frac{x^2 + C}{(1 - x^2)^{1/2}}
   \]

2. **Interpret the Expression**:
   The expression for \( y \) is now written in terms of \( x \) and \( C \). There is no further simplification possible without specific values for \( x \) or \( C \). However, the form of \( y \) depends on \( x \) and the constant \( C \).

This is the general solution to the equation. If you have any specific values for \( x \) or \( C \), they can be substituted into this expression.

Would you like to explore this further or do you have any questions? 

Here are 5 related questions that you might consider:
1. How does the value of \( C \) affect the function \( y(x) \)?
2. What happens to \( y(x) \) as \( x \) approaches 1?
3. How can you interpret the square root in the denominator?
4. What are the domain and range of \( y(x) \) given this function?
5. How would you plot this function \( y(x) \) for different values of \( C \)?

**Tip:** When dealing with expressions that involve square roots and constants, always consider the domain of the function to avoid undefined values.

This problem involves solving an algebraic equation with a square root function and a constant C. Learn the step-by-step process to isolate y and interpret the expression in terms of x and C. Suitable for students in Grades 11-12.

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