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

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Hyperbola with foci $\mathbf{F}_1(-6,-3)$ and $\mathbf{F}_2(4,-3)$, and one vertex $\mathbf{V}_2(3,-3)$.

Key Concepts

Conic Sections

Conic sections are curves that result from the intersection of a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. The shape of the resulting curve depends on the angle and position of the intersecting plane relative to the cone.

Hyperbola

A hyperbola is a type of conic section defined as the set of all points for which the absolute difference in distances to two fixed points, called foci, is constant. This unique property leads to a curve that opens in two opposite directions, usually along a horizontal or vertical axis.

Foci

Foci (singular: focus) are specific fixed points used in the definitions of several conic sections, including hyperbolas. For a hyperbola, the constant difference in distances from any point on the hyperbola to the two foci is what determines its shape and orientation.

Vertices

Vertices are the closest points on a hyperbola to its center and lie along the transverse axis. They act as reference points in standard forms of the hyperbola's equation by defining the value of 'a', which is the distance from the center to a vertex.

Standard Form Equation

The standard form equation of a hyperbola is an algebraic representation that directly relates to its geometric properties, such as the center, vertices, and foci. It is formulated in a way that the distances involved, like those to the vertices and foci, are expressed through parameters, thus linking the geometric and algebraic descriptions of the hyperbola.

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