Questions: Which of the following points satisfy the equation x^2 - xy^2 = 2 ? A. (-√2 / 2, (4.5)^(1 / 4)) B. (0, √2) C. (-√2, 0) D. (0,0) E. None of these.

Solution Steps

To determine which of the given points satisfy the equation \( x^2 - xy^2 = 2 \), we need to substitute each point into the equation and check if the equation holds true.

We need to check which of the following points satisfy the equation \( x^2 - xy^2 = 2 \):

• A. \( \left(-\frac{\sqrt{2}}{2}, (4.5)^{1/4}\right) \)
• B. \( (0, \sqrt{2}) \)
• C. \( (-\sqrt{2}, 0) \)
• D. \( (0, 0) \)

We evaluate the equation for each point:

1. For point A: \[ x = -\frac{\sqrt{2}}{2}, \quad y = (4.5)^{1/4} \implies x^2 - xy^2 = \left(-\frac{\sqrt{2}}{2}\right)^2 - \left(-\frac{\sqrt{2}}{2}\right)(4.5)^{1/2} = \frac{1}{2} + \frac{\sqrt{2}}{2} \cdot \sqrt{4.5} \] This does not equal 2.

2. For point B: \[ x = 0, \quad y = \sqrt{2} \implies x^2 - xy^2 = 0^2 - 0 \cdot (\sqrt{2})^2 = 0 \] This does not equal 2.

3. For point C: \[ x = -\sqrt{2}, \quad y = 0 \implies x^2 - xy^2 = (-\sqrt{2})^2 - (-\sqrt{2}) \cdot 0^2 = 2 \] This equals 2.

4. For point D: \[ x = 0, \quad y = 0 \implies x^2 - xy^2 = 0^2 - 0 \cdot 0^2 = 0 \] This does not equal 2.

From the evaluations, the points that satisfy the equation \( x^2 - xy^2 = 2 \) are:

• Point A: Not valid
• Point B: Not valid
• Point C: Valid
• Point D: Not valid

Final Answer