Finding the Standard Equation of an Ellipse In Exercises...

Finding the Standard Equation of an Ellipse In Exercises $31-36,$ find the standard form of the equation of the ellipse with the given characteristics.
Center: $(1,2)$
Major axis: vertical
Points on the ellipse:
$(1,6),(3,2)$

Key Concepts

Determining Axis Lengths

The lengths of the semi-major axis (a) and semi-minor axis (b) define the shape and size of an ellipse. They can be determined by using the distances from the center to points on the ellipse, such as vertices and co-vertices. Once one of these values is found through geometric reasoning or substitution of known points into the standard equation, the other can often be determined by ensuring the point satisfies the ellipse equation.

Orientation of the Ellipse

The orientation of an ellipse refers to whether its major axis is aligned horizontally or vertically. This orientation determines which variable's term, either (x - h)² or (y - k)², has the larger denominator, corresponding to the square of the semi-major axis. Correct identification of orientation is essential for setting up the proper form of the equation.

Standard Form of an Ellipse

The standard form of the ellipse equation is a way of expressing the ellipse in terms of its center, and the lengths of its semi-axes. It generally takes the form (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center and a and b are the semi-major and semi-minor axes respectively. The form of the equation helps determine key properties such as the orientation and dimensions of the ellipse.

Center of an Ellipse

The center of an ellipse is the point about which the ellipse is symmetric. It serves as the reference point from which the distances to the vertices (along the major axis) and co-vertices (along the minor axis) are measured. In the standard equation, this is represented by the terms (x - h) and (y - k).

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