Latus Rectum

Latus Rectum in Conic Section

A conic section is defined by a second-degree polynomial equation in two variables. Conic sections are classified into four different types namely circle, ellipse, parabola, and hyperbola. The different names are given to the conic section as each conic section is represented by a cross-section of a plane cutting through a cone. To define these curves, many important terms are used such as latus rectum, focus, directrix, etc. In this article, we will discuss the latus rectum of different curves such as a parabola, hyperbola, and ellipse in detail.


What is Latus Rectum?

In the conic section, the word latus rectum is derived from the Latin word “ latus” which means “ side” and the rectum which means “ straight. The latus rectum is defined as  the chord passing through the focus, and perpendicular to the directrix. The end point of the latus rectum lies on the curve.”Half of the latus rectum is considered as the semi latus rectum. The length of the latus rectum of each conic section is defined differently. For example,

  • The length of the latus rectum of the parabola is always equivalent to four times the focal length of the parabola.

  • The length of the latus rectum of an ellipse is defined as the square of the length of the conjugate axis divided by the length of the transverse axis.

  • The length of the latus rectum of a hyperbola is defined as the square of the length of the transverse axis divided by the length of the conjugate axis.

  • The length of the latus rectum of a circle is equivalent to the length of its diameter.


Parabola

A parabola is defined in terms of a line, known as the directrix, and a point not on the directrix is considered as the locus of points that are equidistant from both the directrix and focus. The axis of symmetry is a line that splits the parabola symmetrically. The vertex of the parabola is the intersection of the axis of symmetry and the parabola.

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Latus Rectum of Parabola

The latus rectum of the parabola is a line segment passing through its focus and perpendicular to its axis. The length of the latus rectum of a parabola is always equivalent to four times the distance of the focus from the vertex of the parabola. 


General Equation of Parabola

Let us consider the situation where the axis of the parabola is perpendicular to the y-axis. Accordingly, its equation will be of the type (x - h) = 4a(y-k), where the variables h, a, and k are considered as the real numbers, ( h, k) is its vertex, and 4a is the latus rectum

If we rearrange the formula, we get

x² - 2hx + h² = 4ay - 4ak

4ay = x² - 2hx + h² + 4ak

y = 1/4a ( x² - 2hx + h² + 4ak)

= 1/4a x²  + (- 2h/4a x ) + (h + 4ak/ 4a)

If we replace 1/4a with p,  /4a with q , and h² + 4ak/4a with r , we get the quadratic equation as

 y = px² + qx + r

In the above quadratic equation, p cannot be 0, or else the equation will have a straight line.


Length of the Latus Rectum of Parabola Derivation

Let the end of the latus rectum of a parabola y = 4ax as L and L’. The x - coordinates of L and L’ are equivalent to ‘a’ as S =( a, 0)

Let us consider L = (a,b)

As, we know L is  a point of parabola, we have

b² = 4a (a) = 4a²

Taking square root on both the sides, we get b = 2a

Hence, the end of the latus rectum of parabola are L = ( a, 2a) and L’ =( a, -2a)

Therefore, the length of the latus rectum of parabola LL’ is 4a.


Latus Rectum of Parabola Formula

Latus rectum of parabola formula is 4a, where a is the distance of the focus from the vertex of the parabola.


Hyperbola

A hyperbola is defined as the locus of a point in such a way that the distance to each focus is greater than 1. In other words, the locus of points moving in a plane in such a way that the ratio of its distance from a fixed point i.e. (focus) to its distance from a fixed line ( directrix) is a constant and greater than 1.

Note : The point ( focus ) does not lie on the line (directrix) i.e. (PS/PM) = e > 1 eccentricity.


Latus Rectum of Hyperbola 

The Latus rectum of a hyperbola is defined as a line segment perpendicular to the transverse axis through any of the foci and whose ending point lies on the hyperbola. The length of the latus rectum of a hyperbola is 2b²/4a.


Ellipse

The set of all points in a plane, the sum of whose distance from the fixed point in the plane is constant is an ellipse. The two fixed points in the plane are considered as the foci of the ellipse. When a line segment is formed joining two focus points, the midpoint of this line is considered as the center of an ellipse.. 

The line joining two foci is known as the major axis and a line forms through the center and perpendicular to the major axis is the minor axis. The ending points of the major axis are known as the vertices of the ellipse.

Also, 

  • The length of the major axis of an ellipse is represented by 2a.

  • The length of the minor axis of an ellipse is represented by 2b.

  • The distance of the foci is given by 2c


Latus Rectum of Ellipse

Latus rectum of the ellipse is defined as the length of the line segment perpendicular to the major axis passing through any of the foci and whose endpoint lies on the ellipse. The length of the latus rectum of an ellipse is 2b²/ a.


Latus Rectum Examples

  1. Find the equation of the parabola having its focus ( 0, -3) and the directrix of the parabola is on the line y = 3

Solution: 

As the focus of the parabola is on the y- axis and is also below the directrix, the parabola will be opened downward, and the value of a = -3. Hence, the equation of a parabola is given as x² = 12x. The length of the latus rectum of a parabola is |4 (-3)| = 12.


  1. What will be the equation of hyperbola whose length of the latus rectum is 36 and the foci are ( 0, ±12).

Solution:

Given,

The foci of Hyperbola are (0, ± 12)

That means, the value of C = 12.

Length of the latus rectum of hyperbola = 2b²/a = 36 

or 

b² = 18a

Now, c² = a² + b²

144 = a² + 18a

i.e., a² + 18a – 144 = 0

a = – 24, 6

As the value of a cannot be negative. Therefore, a = 6, and b² = 108.


FAQs (Frequently Asked Questions)

1. What are the different properties of Parabola?

Following are some of the properties of parabola:

  • The eccentricity value of any parabola is 1.

  • The axis of a parabola passes through the vertex and focus, and is also perpendicular to the directrix.

  • The parabola is always symmetric about its axis.

  • The tangent drawn at the vertex of the parabola is parallel to the directrix.

  • The tangents drawn to any point on the directrix of parabola are perpendicular.

2. What are the different properties of an ellipse?

Following are the different properties of an ellipse:

  • The point of intersection of the major axis and minor axis of a parabola is known as the center of the ellipse.

  • The fixed point on the ellipse is known as the focus.

  • The chord of an ellipse is the straight line that passes through two points on the ellipse curve.

  • The major axis is the longest diameter of an ellipse whereas the minor axis is the shortest diameter of an ellipse.

  • The directrix is the fixed straight line of the axis.

  • The tangent is a line that touches the ellipse at just one point.

  • The constant ratio of an ellipse is represented by e i.e. e < 1.

  • The latus rectum of an ellipse is defined as the chord that passes through the focus and perpendicular to the major axis