The equation of a parabola is (y-k)^2=4 a(x-h) where, a>0....

The equation of a parabola is $(y-k)^{2}=4 a(x-h)$ where, $a>0$. The vertex and focus are respectively at (a) $(\mathrm{h}, \mathrm{k})$ and $(\mathrm{a}, \mathrm{k})$ (b) $(\mathrm{h}+\mathrm{a}, \mathrm{k}+\mathrm{a})$ and $(\mathrm{h}, \mathrm{k})$ (c) $(\mathrm{h}, \mathrm{k})$ and $(\mathrm{h}-\mathrm{a}, \mathrm{k})$ (d) $(\mathrm{h}, \mathrm{k})$ and $(\mathrm{h}+\mathrm{a}, \mathrm{k})$

The equation of a parabola is $(y-k)^{2}=4 a(x-h)$ where, $a>0$. The vertex and focus are respectively at
(a) $(\mathrm{h}, \mathrm{k})$ and $(\mathrm{a}, \mathrm{k})$
(b) $(\mathrm{h}+\mathrm{a}, \mathrm{k}+\mathrm{a})$ and $(\mathrm{h}, \mathrm{k})$
(c) $(\mathrm{h}, \mathrm{k})$ and $(\mathrm{h}-\mathrm{a}, \mathrm{k})$
(d) $(\mathrm{h}, \mathrm{k})$ and $(\mathrm{h}+\mathrm{a}, \mathrm{k})$

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Key Concepts

Standard Form of a Parabola

The standard form of a parabola, such as (y - k)^2 = 4a(x - h), is a specific way to write the equation that clearly shows the vertex (h, k) and helps identify other properties of the parabola like its focus and directrix. This form standardizes parabolas to make geometric and algebraic properties more apparent.

Vertex

The vertex of a parabola is the point at which the curve changes direction, representing the minimum or maximum of the function. In the standard form (y - k)^2 = 4a(x - h), the vertex is given by the point (h, k).

Focus

The focus of a parabola is a fixed point used in its geometric definition. Any point on the parabola is equidistant from the focus and a line called the directrix. For a parabola in the form (y - k)^2 = 4a(x - h) with a positive, the focus is located at (h + a, k), indicating the direction in which the parabola opens.

Parameter a

The parameter 'a' in the equation (y - k)^2 = 4a(x - h) determines both the distance between the vertex and the focus and the width or openness of the parabola. A positive 'a' means the parabola opens to the right, while a negative would indicate it opens to the left.

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