Focus of Parabola
Focus of parabola lies on the axis of the parabola. The focus of parabola is helpful in defining the parabola. A parabola represents the locus of a point that is equidistant from a fixed point called the focus, and the fixedline called the directrix. The coordinates of the focus of the parabola depends on the equation of parabola and the axis of the parabola.
Let us learn more about the focus of parabola, and how to find the focus of parabola.
1.  What Is Focus of Parabola? 
2.  How to Locate Focus of Parabola? 
3.  Uses of Focus of Parabola 
4.  Examples on Focus of Parabola 
5.  Practice Questions 
6.  FAQs on Focus of Parabola 
What Is Focus of Parabola?
The focus of a parabola lies on the axis the parabola. The focus of the parabola helps in defining the parabola. A parabola represents the locus of a point which is equidistant from a fixed point called the focus and the fixed line called the directrix. The focus and the directrix are equidistant from the vertex of the parabola. Here we define the focus for the standard equations of a parabola.
 The focus of the parabola y^{2} = 4ax, and having xaxis as its axis is F (a, 0).
 The focus of the parabola y^{2} = 4ax, and having xaxis as its axis is F (a, 0).
 The focus of the parabola x^{2} = 4ay, and having yaxis as its axis is F(0, a).
 The focus of the parabola x^{2} = 4ay and having yaxis as its axis is F(0, a).
How to Locate Focus of Parabola?
The focus of a parabola lies on the axis of the parabola. The focus of a parabola lies at a distance of 'a' units from the vertex of the parabola. The vertex and the focus lies on the axes of the parabola and the axes can be calculated based on the equation of the parabola.
The parabola having an equation with a second degree in x has the yaxis or a line parallel to the yaxis as its axis, and the parabola having an equation with second degree in y had the xaxis or a line parallel to the xaxis as its axis.
Equation of a Parabola  Axis of the Parabola  Vertex of the Parabola  Focus of the Parabola 
(x  h)^{2} = 4a(y  k)  x = h  (h, k)  (h, k + a) 
(y  k)^{2} = 4(x  h)  y = k  (h, k)  (h + a, k) 
Uses of Focus of Parabola
The focus is used to find the numerous features of the parabola.
 The focus helps to write the equation of a parabola.
 The focus of a parabola helps to locate the axis of the parabola.
 The focus of the parabola is useful to find the equations of the focal chords.
 The focus of the parabola is useful to find the length of the latus rectum and the endpoints of the latus rectum.
Examples on Focus of Parabola

Example 1: Find the equation of a parabola having the focus of (4, 0), the xaxis as the axis of the parabola, and the origin as the vertex of the parabola.
Solution:
The given focus of the parabola is (a, 0) = (4, 0)., and a = 4.
For the parabola having the xaxis as the axis and the origin as the vertex, the equation of the parabola is y^{2} = 4ax.
Hence the equation of the parabola is y^{2} = 4(4)x, or y^{2} = 16x.
Therefore, the equation of the parabola is y^{2} = 16x.

Example 2: Find the focus of the parabola having the equation (x  5)^{2} = 24(y  3).
Solution:
The given equation of the parabola is (x  5)^{2} = 24(y  3). The equation resembles the equation of the parabola (x  h)^{2} = 4a(y  k).
The vertex is (h, k) = (5, 3), and 4a = 24, and a = 6.
Hence the focus is (h, k + a) = (5, 3 + 6) = (5, 9).
Therefore, the focus of the parabola is (5, 9).
Practice Questions on Focus of Parabola
Here are a few activities for you to practice. Select your answer and click the "Check Answer" button to see the result.
FAQs on Focus of Parabola
How Do I Find Focus of a Parabola?
The focus of a parabola can be calculated by knowing the axis of the parabola, and the vertex of the parabola. For an equation of the parabola in standard form y^{2} = 4ax, the vertex is the origin and the axis of this parabola is the xaxis. Hence the focus of this parabola is (a, 0). Similarly, we can easily find the focus of the parabola from the equation of a parabola.
What Is the Formula For Focus of Parabola?
There is no defined standard formula for the focus of parabola. For the given equation of the parabola we first need to find the vertex, the value of 'a', and the axis of the parabola, to find the focus of parabola. For a parabola of the form (x  h)^{2} = 4a(y  k), the yaxis is the axis of the parabola, the vertex is (h, k), and the focus of parabola is (h, k + a).
How Are the Focus of Parabola, and Directrix of Parabola Related?
The focus of parabola is a point, and the directrix of parabola is a straight line, which are helpful to define the parabola. A parabola is the locus of a point which is equidistant from a fixed point called the focus, and the fixedline called the directrix. The focus and the directrix lie on either sides of vertex of the parabola and are equidistant from the vertex.
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