Find the focus and directrix of the parabola

If you want to know how to find the focus of a parabola, you first need to define what a parabola is. A parabola is a curved figure where any point on the curve is equal distance from a fixed point (called the focus) and a fixed straight line (called the directrix).

Let’s identify the parts of a parabolic function. In the graph above, you see a given line that intersects the directrix at a 90-degree angle. This straight line is called the axis of symmetry. The point that is marked C, signifying where the parabola opens, is called the vertex. The vertex is always midway between the focus and directrix of a parabola.

The Equation of a Parabola

The above graph is a basic representation of a parabola where the coordinates of the vertex are (0,0). When you draw the axis of symmetry through the parabola's vertex, you see that this vertical line perfectly matches up with the y-axis of the graph. This parabola is represented by equation

If this parabola was rotated 90 degrees to the right, the fixed line representing the axis of symmetry would be situated along the x-axis. The equation of the parabola is now

The standard form of the equation of a parabola, where the conic shape of the parabola is formed along the y-axis, is

How To Find the Focus of a Parabola

In order to find the focus of a parabola, you must know that the equation of a parabola in a vertex form is y=a(x−h)2+k where a represents the slope of the equation. From the formula, we can see that the coordinates for the focus of the parabola is (h, k+1/4a).

So now, let's solve for the focus of the parabola below:

When we use the above coordinates, the equation of the parabola above is

We've determined that the points of the focus are (0,2).

Let’s Review How To Find the Focus of a Parabola

In order to determine how to find the focus of a parabola, you 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 you to expand upon your basic understanding of graphing geometric equations.

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• How To Find the Axis of Symmetry of a Parabola

Directrix of parabola is perpendicular to the axis of the parabola. The directrix of the 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 eqution of the directrix of parabola depends on the equation of the parabola, the focus, and the axis of the parabola.

Let us learn more about the directrix of parabola, and how to find the directrix of parabola.

What Is Directrix of Parabola?

The directrix of a parabola is a line that is perpendicular to the axis of the parabola. The directrix 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 directrix and the focus are equidistant from the vertex of the parabola. Here we define the directrix for the standard equations of a parabola.

• The directrix of the parabola y2 = 4ax, having x-axis as its axis, passes through (-a, 0), and has the equation x + a = 0.
• The directrix of the parabola y2 = -4ax, having x-axis, passes through (a, 0), and has the equation x - a = 0.
• The directrix of the parabola x2 = 4ay, having y-axis as its axis, passes through (0, -a), and has the equation y + a = 0.
• The focus of the parabola x2 = -4ay, having y-axis as its axis, passes through (0, a), and has the equation y - a = 0.

How to Locate Directrix of Parabola?

The directrix of a parabola is a line perpendicular to the axis of the parabola. The directrix of a parabola lies at a distance of 'a' units from the vertex of the parabola. The directrix and the focus lie at equal distance from the vertex of the parabola and the equation of directrix can be calculated based on the equation of the parabola.

The parabola having an equation with a second degree in x has the y-axis or a line parallel to the y-axis as its axis, and the parabola having an equation with second degree in y had the x-axis or a line parallel to the x-axis as its axis.

Uses of Directrix of Parabola

The directrix of a parabola is used to find the numerous features of the parabola.

• The directrix of a parabola helps to write the equation of a parabola.
• The directrix of a parabola helps to locate the axis of the parabola.
• The directrix of the parabola is useful to find the equations of the focal chords.
• The directrix of the parabola is useful to find the equation of the latus rectum and the endpoints of the latus rectum.

Examples on Directrix of Parabola

1. Example 1: Find the equation of a parabola having the directrix of parabola as x + 5 = 0, the x-axis as the axis of the parabola, and the origin as the vertex of the parabola.

   Solution:

   The directrix of parabola is x + 5 = 0. The focus of the parabola is (a, 0) = (5, 0).

   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(5)x, or y2 = 20x.

   Therefore, the equation of the parabola is y2 = 20x.

2. Example 2: Find the directrix of parabola, having the equation (y - 7)2 = 12(x - 4).

   Solution:

   The given equation of the parabola is (y - 7)2 = 12(x - 4). The equation resembles the equation of the parabola (y - k)2 = 4a(x - h).

   The vertex is (h, k) = (4, 7), and 4a = 12, and a = 3.

   Hence the focus is (h + a, k ) = (4 + 3, 7) = (7, 7).

   The directrix is passing through (h - a, k) = (4 - 3, 7) = (1, 7)

   The equation of the directrix of parabola is x = 1, or x - 1 = 0

   Therefore, the equation of directrix of parabola is x - 1 = 0.

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FAQs on Directrix of Parabola

How Do I Find Directrix of a Parabola?

The directrix of a parabola can be found, by knowing the axis of the parabola, and the vertex of the parabola. For an equation of the parabola in standard form y2 = 4ax, with focus at (a, 0), axis as the x-axis, the equation of the directrix of this parabola is x + a = 0 . Similarly, we can easily find the directrix of the parabola for the other forms of equations of a parabola.

What Is the Equation For Directricx of Parabola?

The equation of directrix is the line that is perpendicular to the axis of the parabola. For the given equation of the parabola we first need to find the vertex, focus, and axis of the parabola, to find the equation of directrix of the parabola. For a parabola of the form (x - h)2 = 4a(y - k), the line parallel to the y-axis is the axis of the parabola, the vertex is (h, k), focus of parabola is (h, k + a), and the equation of directrix of parabola is y = k - a

How Are the Focus of Parabola, and Directrix of Parabola Related?

The focus of parabola is a point, and the directrix of the parabola is a straight line, which is together helpful to define the equation of a parabola. A parabola is 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 lie on either side of the vertex of the parabola and are equidistant from the vertex.

How do you find the focus of a parabola?

How do I find the Directrix of a parabola?

How do you find the focus and Directrix and focal diameter of a parabola?