Find the equation of the ellipse which has vertices V1(-2,6), V^2(-2,-4), and foci F1(-2,4), F2(

step 1 The ellipse with vertices is v1 is equal to minus 1 minus 4 and v2 is equal to minus 1 and 6 and

Find the equation of the ellipse which has vertices...

Key Concepts

Orientation of the Ellipse

The orientation of an ellipse refers to the direction of its major axis. Depending on whether the vertices are aligned horizontally or vertically in the coordinate plane, the ellipse is classified as horizontally or vertically oriented, which in turn affects the form of its standard equation.

Ellipse

An ellipse is a type of conic section defined as the set of all points in a plane such that the sum of the distances from these points to two fixed points (called foci) is constant. This geometric definition leads to important properties regarding its shape, symmetry, and the relationship between its axes and foci.

Center of the Ellipse

The center of an ellipse is the midpoint of the line segment joining the vertices. It represents the point about which the ellipse is symmetric and is used to translate the ellipse to a standard position in the coordinate plane.

Vertices

Vertices are the points where the ellipse intersects its major axis, representing the farthest points from the center along that axis. They determine the length of the major axis and help in locating the ellipse’s orientation in the coordinate plane.

Foci

The foci (plural of focus) are two specific points inside the ellipse used in its definition. The distance from any point on the ellipse to each focus, when added together, remains constant. The relative position of the foci compared to the vertices helps determine the eccentricity of the ellipse.

Standard Equation of an Ellipse

The standard form of an ellipse’s equation, centered at (h, k), is written as ((x-h)^2)/(a^2) + ((y-k)^2)/(b^2) = 1 for a horizontally oriented ellipse, or ((x-h)^2)/(b^2) + ((y-k)^2)/(a^2) = 1 for a vertically oriented ellipse, where a and b represent the semi-major and semi-minor axes respectively.

Relationship Between a, b, and c

In an ellipse, the relationship between the semi-major axis (a), the semi-minor axis (b), and the focal distance (c) is given by the equation c^2 = a^2 - b^2 (if the major axis is horizontal) or b^2 = a^2 - c^2 (if the major axis is vertical), linking the geometric configuration of the ellipse with its algebraic representation.

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