Consider the parabola 4 x^2-4 x+8 y+17=0 (a) latus rectum of...

Consider the parabola $4 x^{2}-4 x+8 y+17=0$
(a) latus rectum of the parabola is 2
(b) no part of the curve lies above the line $y+2=0$
(c) focus of the parabola is at $\left(\frac{1}{2}, \frac{-3}{2}\right)$
(d) focus of the parabola is at $\left(\frac{1}{2}, \frac{-5}{2}\right)$

Key Concepts

Focus of a Parabola

The focus is a fixed point used in the definition of a parabola; every point on the parabola is equidistant from the focus and a corresponding directrix. Its position relative to the vertex determines the parabola's opening direction and is calculated using the parameter p from the vertex form.

Latus Rectum

The latus rectum is a chord of the parabola that passes through the focus and is parallel to the directrix. Its length, which is 4|p|, is an important geometric feature that provides insight into the ‘width’ of the parabola at the focus.

Completing the Square

This is an algebraic method used to convert a quadratic equation into its vertex form. By reorganizing and grouping terms, one can rewrite the equation of a parabola into a form that clearly reveals its vertex, and from which other properties such as the focus, directrix, and orientation can be easily identified.

Vertex Form of a Parabola

The vertex form of a parabola expresses the equation in the format (x-h)² = 4p(y-k) or (y-k)² = 4p(x-h), where (h, k) is the vertex and p is a parameter indicating the distance from the vertex to the focus or directrix. This form makes it straightforward to extract geometric properties of the parabola.

Orientation of a Parabola

This concept refers to the direction in which the parabola opens (upward, downward, leftward, or rightward). The orientation is determined by the structure of the vertex form of the parabola and by the sign and position of the parameter p. It helps in understanding the spatial arrangement of the parabola with respect to given lines or axes.

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