Question: If the center of the locket is at the origin of a coordinate plane, which equation represents the…

If the center of the locket is at the origin of a coordinate plane, which equation represents the locket?

\[ \frac{y^2}{361} + \frac{x^2}{169} = 1 \]

\[ \frac{x^2}{1,444} + \frac{y^2}{676} = 1 \]

\[ \frac{y^2}{1,444} + \frac{x^2}{676} = 1 \]

\[ \frac{x^2}{361} + \frac{y^2}{169} = 1 \]

Solution

To determine which equation represents the locket, we must recognize the standard form of the equation for an ellipse centered at the origin on a coordinate plane: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] or \[ \frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 \] The question provides a choice of equations resembling the general form of an ellipse. We need to ensure that the equation matches the form where the sum of the fractions equals 1, and variables are squared. The options are: 1. \(\frac{y^2}{361} + \frac{x^2}{169} = 1\) 2. \(\frac{x^2}{1,444} + \frac{y^2}{676} = 1\) 3. \(\frac{y^2}{1,444} + \frac{x^2}{676} = 1\) 4. \(\frac{x^2}{361} + \frac{y^2}{169} = 1\) Since the center is at the origin, the correct format should include both squared variables and divide by constants yielding 1. The correct equation that adheres to this form for an ellipse centered at the origin should be: \[ \frac{x^2}{361} + \frac{y^2}{169} = 1 \] This matches option 4.