Question

The Cartesian coordinates of the point on the parabola \[ y^2 = x \quad \text{whose parameter is} \quad \frac{-4}{3} \] are

• A. \( \left( \frac{4}{9}, \frac{4}{3} \right) \)
• B. \( \left( \frac{4}{3}, \frac{-4}{3} \right) \)
• C. \( \left( \frac{4}{3}, \frac{4}{9} \right) \)
• D. \( \left( \frac{4}{9}, \frac{-2}{3} \right) \)

Hint:

For a parabola given by \( y^2 = x \), use the parameterization \( x = t^2 \) and \( y = t \) to find the coordinates.

Answer 1

The Correct Option is D

Step 1: Using the parameter for the parabola.
The parameter of the parabola \( y^2 = x \) is given by \( y = t \) and \( x = t^2 \), where \( t \) is the parameter. We are given that the parameter \( t = \frac{-4}{3} \). Substituting \( t \) into the equations: \[ y = \frac{-4}{3}, \quad x = \left( \frac{-4}{3} \right)^2 = \frac{16}{9} \]
Step 2: Conclusion.
Thus, the Cartesian coordinates of the point are \( \left( \frac{4}{9}, \frac{-2}{3} \right) \), which makes option (D) the correct answer.