Find a Cartesian equation for the conic section satisfying...

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Ellipse with foci $\mathbf{F}_1(-3,2)$ and $\mathbf{F}_2^{\prime}(-3,6)$, and major axis of length 12.

Key Concepts

Standard Form of the Ellipse Equation

The standard form equation of an ellipse expresses the relationship between the coordinates of any point on the ellipse and the lengths of its semi-major and semi-minor axes. Depending on whether the major axis is horizontal or vertical, this equation typically appears as (x - h)²/b² + (y - k)²/a² = 1 or (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center of the ellipse.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double?napped cone. This family includes ellipses, parabolas, and hyperbolas, and each type is defined by specific geometric properties. An ellipse, in particular, is formed when the intersection produces a closed, symmetric curve with distinct axes.

Ellipse

An ellipse is defined as the set of points in a plane for which the sum of the distances to two fixed points, known as the foci, is constant. This constant is the length of the major axis, and it determines the overall size and shape of the ellipse.

Foci

The foci are two fixed points inside an ellipse that are crucial to its definition. The key property of an ellipse is that the sum of the distances from any point on the ellipse to these two foci remains constant, which governs the shape and geometry of the curve.

Major Axis

The major axis is the longest line segment that passes through the center of the ellipse and connects two points on the ellipse. Its length, being twice the semi-major axis length, is equal to the constant sum of distances from any point on the ellipse to the foci.

Center of an Ellipse

The center of an ellipse is the midpoint of the line segment joining its two foci. This point serves as the origin for translating the ellipse to its standard position, and it is the point of symmetry that simplifies the representation of the ellipse in Cartesian coordinates.

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