Find the standard form of the equation of each ellipse satisfying the given conditions. Major ax

step 1 Hello, we have problem number 34. In this we have major xes vertical with length 20. We have to

Find the standard form of the equation of each ellipse...

Key Concepts

Ellipse Center Translation

This concept involves shifting the graph of the ellipse so that its center is not at the origin, but at a point (h, k). In the standard form of an ellipse's equation, (x - h) and (y - k) replace x and y respectively, to account for this translation. This shift is vital in accurately representing the ellipse's position in the plane.

Semi-Axes Length

The semi-major and semi-minor axes are half the lengths of the ellipse's major and minor axes. They determine the size and shape of the ellipse. Knowing these lengths is essential as they form the denominators in the standard form of the ellipse's equation, influencing the curvature in the horizontal and vertical directions.

Ellipse Orientation

The orientation of the ellipse is determined by the direction of its major axis. If the major axis is vertical, then the larger denominator in the ellipse's equation corresponds to the vertical component. This information is used to correctly set up the standard form of the ellipse's equation, ensuring that the roles of the semi-major and semi-minor axes are appropriately assigned.

Standard Equation of an Ellipse

The standard form of an ellipse's equation is a mathematical representation that incorporates the center coordinates, semi-major axis, and semi-minor axis. For an ellipse with center (h, k), a vertical major axis uses the form ((x - h)²)/(b²) + ((y - k)²)/(a²) = 1, where a and b are the semi-major and semi-minor axes respectively. This form ensures the geometric properties of the ellipse are maintained.

Please give Ace some feedback