Focus of a parabola

Introduction to Parabola & Focus of Parabola

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".
Parabolas vary in direction and shape. The lowest or highest point in a parabola is called a vertex, which lies on the axis of symmetry. If the leading coefficient of the term to the second degree is positive, the parabola faces up. If it is negative, the parabola faces down.

The name "parabola" is derived from a New Latin term that means something similar to "compare" or "balance", and refers to the fact that the distance from the parabola to the focus is always equal to (that is, is always in balance with) the distance from the parabola to the directrix.

The "general" form of a parabola's equation is the one you're used to, y = ax2 + bx + c — unless the quadratic is "sideways", in which case the equation will look something like x = ay2 + by + c. The important difference in the two equations is in which variable is squared: for regular (vertical) parabolas, the x part is squared; for sideways (horizontal) parabolas, the y part is squared.
The "vertex" form of a parabola with its vertex at (h, k) is:

Regular: y = a(x – h) 2 + k

Sideways: x = a(y – k) 2 + h                                                                 

Equations of Parabola Focus And Directrix

If a parabola has a horizontal axis, the standard form of the equation of the parabola is this: (y - k) 2 = 4p(x - h), where p≠ 0. The vertex of this parabola is at (h, k). The focus is at (h + p, k). The directrix is the line x = h - p.

The axis of the parabola is y-axis. Equation of directrix is y = -a. i.e. y = -½ is the equation of directrix. Vertex of the parabola is (0, 0).  

How to Locate Focus of Parabola? (Explain with Solved Examples)

In order to find the focus of a parabola, individuals must know that the equation of a parabola in a vertex form is y=a (x−h) 2+k where a, demonstrates or represents the slope of the equation. From the formula, it could be seen that the coordinates for the focus of the parabola is (h, k+1/4a). In order to determine and consider how to find the focus of a parabola, students must identify the vertex and plug it into the equation of a parabola. Understanding different properties of a parabola, like the axis of symmetry, directrix, and reflectors, allows students to expand upon their basic understanding of graphing geometric equations.                                                                                                                                                      

Uses Of 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 fixed-line called the directrix. The coordinates of the focus of the parabola depends on the Equation of parabola and the axis of the parabola. The focus of the parabola helps in defining the parabola. In other words, 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.

• The focus of the parabola y2 = 4ax, and having x-axis as its axis is F (a, 0).
• The focus of the parabola y2 = -4ax, and having x-axis as its axis is F (-a, 0).
• The focus of the parabola x2 = 4ay, and having y-axis as its axis is F (0, a).
• The focus of the parabola x2 = -4ay and having y-axis as its axis is F (0, -a).                                                                                   

The focus is used to find the numerous features of the parabola.

1. The focus helps to write the equation of a parabola.
2. The focus of a parabola helps to locate the axis of the parabola.
3. The focus of the parabola is useful to find the equations of the focal chords.
4. 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 (Explain with 3 Solved Examples)                                                                                                                    

Example 1:

Find the focus of the parabola having the equation (x - 5)2 = 24(y - 3).

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).

Example 2:

Find the equation of a parabola having the focus of (4, 0), the x-axis as the axis of the parabola, and the origin as the vertex of the parabola.

The given focus of the parabola is (a, 0) = (4, 0). and a = 4.

For the parabola having the x-axis as the axis and the origin as the vertex, the equation of the parabola is y2 = 4ax.

Hence the equation of the parabola is y2 = 4(4) x, or y2 = 16x.

Therefore, the equation of the parabola is y2 = 16x.

Example 3:

Find the equation of the parabola with vertex at (0, 0) and focus at (0, 4).

Since the vertex is at (0, 0) and the focus is at (0, 5) which lies on y-axis, the y-axis is the axis of the parabola. 

Hence, the equation of the parabola is x2= 4ay. 

Hence, we have x2 = 4(4) y, i.e. 

X2 = 16y