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.
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.
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.
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.
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
Find the equation of the parabola having its focus ( 0, -3) and the directrix of the parabola is on the line y = 3
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.
What will be the equation of hyperbola whose length of the latus rectum is 36 and the foci are ( 0, ±12).
The foci of Hyperbola are (0, ± 12)
That means, the value of C = 12.
Length of the latus rectum of hyperbola = 2b²/a = 36
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.