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The general equation of a parabola is: y = a (x-h) 2 + k or x = a (y-k) 2 +h, where (h,k) denotes the vertex. The standard equation of a regular parabola is y 2 = 4ax. Some of the important terms below are helpful to understand the features and parts of a parabola y 2 = 4ax.
- Focus
Example 1: Find the equation of a parabola having the focus...
- Eccentricity of Parabola
Example 2: Determine the eccentricity of parabola if a = 8...
- Locus
The above equation can be converted to the form \(\dfrac{x^2...
- Focus
Given the focus (h,k) and the directrix y=mx+b, the equation for a parabola is (y - mx - b)^2 / (m^2 +1) = (x - h)^2 + (y - k)^2. Equivalently, you could put it in general form: x^2 + 2mxy + m^2 y^2 -2[h(m^2 - 1) +mb]x -2[k(m^2 + 1)^2 -b]y + (h^2 + k^2)(m^2 + 1) - b^2 = 0
A parabola is defined as the locus (or collection) of points equidistant from a given point (the focus) and a given line (the directrix). Another important point is the vertex or turning point of the parabola. If the equation of a parabola is given in standard form then the vertex will be \((h, k) .\)
f(x) = ax2 + bx + c. where a, b, and c are real numbers and a ≠ 0. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.
The parabola is the curve formed from all the points (x, y) that are equidistant from the directrix and the focus. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola up the middle) is called the axis of symmetry.
The equations of parabolas with vertex (0, 0) (0, 0) are y 2 = 4 p x y 2 = 4 p x when the x-axis is the axis of symmetry and x 2 = 4 p y x 2 = 4 p y when the y-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key features.
How to find axis of symmetry of a parabola. For quadratic equations in general form ax 2 + bx + c, the axis of symmetry can be found using the equation . To find the y-coordinate of the vertex, find the axis of symmetry and substitute that x-value into the original equation.
