SUBMIT ANSWER 9. [-/1 Points] DETAILS MY NOTES LARPCALC10 10.3.011. Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: \( ( \pm 7,0) \); foci: \( ( \pm 4,0) \) Need Help? ReadII SUBMIT ANSWER

To find the standard form of the equation of an ellipse centered at the origin, we start with the information given about the vertices and foci. Since the vertices are at \( (\pm 7, 0) \), we can identify \( a = 7 \). The foci at \( (\pm 4, 0) \) give us \( c = 4 \). We use the relationship \( c^2 = a^2 - b^2 \) to find \( b \), where \( b \) is the semi-minor axis. So, we have: - \( a^2 = 7^2 = 49 \) - \( c^2 = 4^2 = 16 \) Now we solve for \( b^2 \): \[ c^2 = a^2 - b^2 \implies 16 = 49 - b^2 \implies b^2 = 49 - 16 = 33 \] Therefore, the standard form of the equation of the ellipse is: \[ \frac{x^2}{49} + \frac{y^2}{33} = 1 \] This equation reflects the specific dimensions and orientation of the ellipse, ensuring the vertices and foci are correctly placed along the x-axis!