Is Directrix always perpendicular to axis?

A parabola is the open curve formed by the intersection of a plane and a right circular cone. It occurs when the plane is parallel to one of the generatrices of the cone (Figure 1).

A parabola can also be defined as the set of points which are equidistant from a fixed point (the "focus") and a fixed line (the "directrix") (Figure 2).

A third definition is the set of points (x,y) on the coordinate plane which satisfy an equation of the form y = x2, or, more usefully, 4ky = x2. Other forms of equation are possible, but these are the simplest.

The "axis" of a parabola is the line which passes through the focus and is perpendicular to the directrix. The "vertex" is the point where the axis crosses the parabola. The "latus rectum" is the chord passing through the focus and perpendicular to the axis. Its length is four times the distance from the focus to the vertex.

When a parabola is described by the equation 4ky = x2, the vertex is at the origin; the focus is at (o,k); the axis is the y-axis; the directrix is the line y = -k.

In spite of the infinitude of cones—from skinny ones to fat ones—that yield parabolas, all parabolas are geometrically similar. If one has two parabolas, one of them can always be enlarged, as with a photographic enlarger, so that it exactly matches the other. This can be shown algebraically with an example. If y = x2 and y = 3x2 are two parabolas, the transformation x = 3xy = 3y which enlarges a figure to three times its original size, transforms y = x2 into 3y = (3x)2, which can be simplified to y = 3x2.

This reflects the fact that all parabolas have the same eccentricity, namely 1. The eccentricity of a conic section is the ratio of the distances point-to-focus divided by point-to-directrix, which is the same for all the points on the conic section. Since, for a parabola, these two distances are always equal, their ratio is always 1.

A parabola can be thought of as a kind of limiting shape for an ellipse, as its eccentricity approaches 1. Many of the properties of ellipses are shared, with slight modifications, by parabolas. One such property is the way in which a line intersects it. In the case of an ellipse, any line which intersects it and is not simply tangent to it, intersects it in two points. So, surprisingly, does a line intersecting a parabola, with one exception. A line which is parallel to the parabola's axis will intersect in a single point, but if it misses being parallel by any amount, however small, it will intersect the parabola a second time. The parabola continues to widen as it leaves the vertex, but it does so in this curious way.

A parabola's shape is responsible for another curious property. If one draws a tangent to a parabola at any point P, a line FP from the focus to P and a line XP parallel to the axis, will make equal angles with the tangent. In Figure 3, - FPA =- XPB. This means that a ray of light parallel to the axis of a parabola would be reflected (if the parabola were reflective) through the focus, or a ray of light, originating at the focus, would be reflected along a line parallel to the axis.

A parabola, being an open curve, does not enclose an area. If one draws a chord between two points on the parabola, however, the parabolic segment formed does have an area, and this area is given by a remarkable formula discovered by Archimedes in the third century B.C. In Figure 4, M is the midpoint of the chord AB. C is the point where a line through M and parallel to the axis intersects the parabola. The area of the parabolic segment is 4/3 times the area of triangle ABC. For example, the area of the parabola y = x2 and the line y = 9 is (4/3)(6 × 9/2) or 36. What is particularly remarkable about this formula is that it does not involve the number π as the formulas for the areas of circles and ellipses do.

In mathematics, a parabola is a set of all points in a plane that are an equal distance away from a given point and given line. The point is called the focus of the parabola, and the line is called the directrix of a parabola. In other words, parabolas are commonly known as the graphs of quadratic functions. They can also be viewed as the set of all points whose distance from a certain point (the focus) is equal to their distance from a certain line (the directrix).

We will learn about the definition of the directrix of a parabola, the equation of the parabola with focus and directrix, the relationship between the focus and the directrix of a parabola, and the uses of the directrix of a parabola with solved examples to understand the topic deeply.

What is Directrix of a Parabola?

Definition of directrix of a parabola:  Directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola and does not touch the parabola. If the axis of symmetry of a parabola is vertical, the directrix is a horizontal line. The directrix of a parabola helps in defining the parabola.

Focus and Directrix of a Parabola Standard Equation

Given below we defined the directrix, focus, and equation of axis for the standard equations of a parabola:

1. Standard equation when Parabola opening to the Right (i.e. \(y^{2} = 4ax, a > 0\))

• Focus is \(S(a, 0)\).
• Equation of the directrix \(MZ\) is \(x + a = 0\).
• Equation of the axis is \(y = 0\) i.e. \(X\)-axis.

2. Standard equation when Parabola opening to the Left (i.e., \(y^{2} = – 4ax, a > 0\))

• Focus is \(S(-a, 0)\).
• Equation of the directrix \(MZ\) is \(x – a = 0\).
• Equation of axis is \(y = 0\) i.e. \(X\)-axis.

3. Standard equation when Parabola opening Upwards (i.e., \(x^{2} = 4ay, a > 0\))

• Focus is \(S(0, a)\).
• Equation of the directrix \(MZ\) is \(y + a = 0\).
• Equation of the axis is \(x = 0\) i.e. \(Y\)-axis.

4. Standard equation when Parabola opening Downwards (i.e., \(x^{2} = – 4ay, a > 0\))

• Focus is \(S(0, -a)\).
• Equation of the directrix \(MZ\) is \(y – a = 0\).
• Equation of axis is \(x = 0\) i.e. \(Y-axis\).

How to Find the Equation of a Parabola?

We can easily find the equation of a parabola if the focus and directrix of a parabola are given by using the below steps:

Step 1: Determine the orientation of the parabola using the directrix of a parabola.

• If the equation of the directrix of a parabola is of the form \(y = b\), for some number \(b\), then the directrix of a parabola is horizontal and the parabola is vertical. If the focus of a parabola is above the directrix, then the parabola opens upward. If the focus of a parabola is below the directrix, then the parabola opens downward.

• If the equation of the directrix of a parabola is of the form \(x = a\), for some number \(a\), then the directrix of a parabola is vertical and the parabola is horizontal. If the focus of a parabola is to the right of the directrix, then the parabola opens to the right. If the focus of a parabola is to the left of the directrix, then the parabola opens to the left.

Step 2: Using the orientation of the parabola determined in Step 1 to write the correct form of the equation of a parabola.

• If the parabola is vertical, then the equation of a parabola is of the form \((x-h)^{2}=4p(y-k)\).

• If the parabola is horizontal, then the equation of a parabola is of the form \((y-k)^{2}=4p(x-h)\).

Here, \((h, k)\) is the midpoint between the focus and the directrix (i.e. the vertex), and \(|p|\) is half the distance between the focus and the directrix of a parabola. If the parabola opens upward or to the right, then \(p>0\) and vice-versa.

Step 3: Using the coordinates of the focus \((x_{f},y_{f})\) and the value of \(a\) or \(b\) from Step 1 to fill in for \(h\), \(k\) and \(p\) in the equation from Step 2.

• If the parabola is vertical, then \(h=x_{f}\), \(k=\frac{y_{f}+b}{2}\), and \(|p|=\frac{|y_{f}-b|}{2}\).
• If the parabola is horizontal, then \(h=\frac{x_{f}+a}{2}\), \(k=y_{f}\), and \(|p|=\frac{|x_{f}-a|}{2}\).

Example to determine the equation of a parabola with focus \((3, -1)\) and directrix \(x=6\).

Solution: Given that focus of a parabola is \((3, -1)\) and the directrix of a parabola is \(x=6\).

Step 1: The parabola is horizontal and opens to the left, meaning \(p<0\).

Step 2: The equation of a parabola is of the form \((y-k)^{2}=4p(x-h)\).

Step 3: The vertex \((h, k)\) is the midpoint of \((3, -1)\) and the point \((6, -1)\) on the directrix.

\((h, k) = \left(\frac{3+6}{2},-1\right) = \left(\frac{9}{2},-1\right)\).

Half the distance from \((3, -1)\) to \((6, -1)\) is

\(|p| = \frac{|3-6|}{2} = \frac{3}{2}\). Because \(p<0\) \(\Rightarrow\)  \(p = -\frac{3}{2}\).

Substitute all this information into the equation, then

\((y-(-1))^{2} = 4\left(-\frac{3}{2}\right)\left(x-\frac{9}{2}\right)\)

\((y+1)^{2} = -6\left(x-\frac{9}{2}\right)\).

Relationship Between Focus and Directrix of a Parabola

The focus of the parabola is a point, and the directrix of the parabola is a straight line, which together helps to define the equation of a parabola. A parabola is the locus of a point that 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.

Relationship between focus and directrix of a parabola is that the distance of every point on the parabola curve from its focus point and from its directrix is always the same.

Application of Directrix of a Parabola

Directrix of a parabola is applied in various features of the parabola. Some of their applications are given as follows:

1. The directrix of a parabola is used to locate the axis of the parabola.
2. The directrix of a parabola is used to write the equation of a parabola.
3. The directrix of the parabola helps to find the equations of the focal chords.
4. The directrix of the parabola helps to find the equation of the latus rectum and the endpoints of the latus rectum.

Directrix of a Parabola Examples

Example 1: Find the equation of a parabola having the directrix of a parabola as \(x+7 = 0\). The \(x\)-axis is the axis of the parabola, and the origin is the vertex of the parabola.

Solution: The directrix of a parabola is \(x+7 = 0\).

The focus of the parabola is \((a, 0)\) = \((7, 0)\).

For the parabola having \(x\)-axis 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 \(y^{2} = 4(7)x = 28x\).

Therefore, the equation of parabola is \(y^{2} = 28x\).

Example 2: Find the equation of the parabola whose focus is \((4, 0)\) and the directrix is \(x = – 3\).

Solution: Given that the focus lies on the \(x\)-axis, the \(x\) -axis itself is the axis of the parabola.

Hence, the equation of the parabola is in the form either

\(y^{2} = 4ax\) or \(y^{2} = – 4ax\).

Since the equation is \(x = – 3\) and the focus is \((4, 0)\),

Then the parabola is to be of the form \(y^{2} = 4ax\) with \(a = 4\).

Hence the required equation is \(y^{2} = 4(4)x\)

\(y^{2} = 16x\).

Example 3: Find the focus and directrix of a parabola: \(y^{2} = 8x\).

Solution: The given parabola is of the form \(y^{2} = 4ax\), where

\(4a = 8\),

\(a = 2\)

The coordinates of the focus are \((a, 0)\), i.e. \((2, 0)\)

and, the equation of directrix is

\(x = – a\), i.e. \(x = – 2\).

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Directrix of a Parabola FAQs

Ans.1 Yes, the directrix has the property of being always perpendicular to the axis of symmetry of the parabola

Ans.2 The focus and directrix of a parabola is found by knowing the axis of the parabola, and the vertex of the parabola. For an equation of the form \(y^{2} = 4ax\), the focus is \((a, 0)\), axis as the \(x\)-axis, the equation of the directrix of this parabola is \(x + a = 0\). Similarly, find the focus and directrix of the parabola for the other forms of equations of a parabola.

Ans.3 Directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola and does not touch the parabola.

Ans.4 1. Standard equation when Parabola opening to the Right (i.e. \(y^{2} = 4ax, a > 0\)) 2. Standard equation when Parabola opening to the Left (i.e., \(y^{2} = – 4ax, a > 0\)) 3. Standard equation when Parabola opening Upwards (i.e., \(x^{2} = 4ay, a > 0\))

4. Standard equation when Parabola opening Downwards (i.e., \(x^{2} = – 4ay, a > 0\))

Ans.5 Relationship between focus and directrix of a parabola is that the distance of every point on the parabola curve from its focus point and from its directrix is always the same.