The equation of the parabola whose vertex is at (2,1) and...

The equation of the parabola whose vertex is at $(2,1)$ and whose focus is at $(1,-1)$ is
(a) $4 x^{2}-4 x y+y^{2}-32 x-34 y+89=0$
(b) $4 x^{2}-4 x y+y^{2}+8 x+46 y-71=0$
(c) $4 x^{2}+4 x y+y^{2}-32 x-34 y+89=0$
(d) $4 x^{2}+4 x y+y^{2}+8 x+46 y-71=0$

Key Concepts

Parabola as a Conic Section

A parabola is defined as the set of all points in the plane that are equidistant from a fixed point, called the focus, and a fixed line, known as the directrix. This definition underpins the derivation of its equation and explains why the distance to the focus equals the distance to the directrix for any point on the curve.

Vertex and Focus Relationship

The vertex of a parabola is significant because it is the point where the parabola makes its sharpest turn and is equidistant from the focus and the directrix. Knowing the vertex and one of these elements (commonly the focus) allows for the determination of the second (the directrix) and thus the full derivation of the parabola's equation.

General Second-Degree Equation and Discriminant

Conic sections, including parabolas, can be expressed by a general second-degree equation in x and y. For a parabola, the discriminant term (B² - 4AC) equals zero, which is a key condition that helps distinguish it from ellipses and hyperbolas. This aspect is particularly important when the equation includes an xy term, indicating a rotated conic.

Rotation of Axes in Conic Sections

When a conic section, such as a parabola, is rotated, the standard form of its equation changes to include an xy term. Understanding how rotation affects the equation is crucial for rewriting the equation in a form that clearly reveals the conic's geometric properties, such as its vertex, focus, and axis of symmetry.

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