Latus Rectum of Conic Section Introduction
In Mathematics, a conic section is represented as a curve which we get from the intersection of the surface of a cone. There are varied types of conic sections. These are parabola, ellipse, and hyperbola. To represent these curves, many important terms are used such as focus, directrix, latus rectum, locus, asymptote, etc. In this article, we will study about latus rectum, latus rectum definitions, latus rectum examples, and latus rectum of conic sections.
The latus rectum of the conic section is stated as the chord that passes through the focus and is perpendicular to the major axis and includes both endpoints on the curve.
The length of the latus rectum is specified differently for each conic section:
The length of the latus rectum in a circle is always equivalent to the length of the diameter in a circle.
The length of the latus rectum in a parabola is equal to four times the focal length.
The length of the latus rectum in a hyperbola is equal to twice the square of the length of the transverse axis divided by the length of the conjugate axis.
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Latus Rectum Definition
In the conic section, the latus rectum is the chord drawn through the focus and parallel to the directrix. The word latus is derived from the Latin word “ latus’’ which implies side and the term “rectum” meaning straight. Half of the latus rectum is known as the semi latus rectum. The diagram given below represents the latus rectum of a parabola.
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Length of the Latus Rectum of a Parabola
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Let us consider the length of the latus rectum of the parabola y2= 4ax as L and L’. The x coordinates of L and L’ are equivalent to “a” as S = (a, 0)
Let us assume L = (a, b)
As we know, L is the point of parabola. Accordingly, we have
b2 = 4a (a) = 4a2
Taking square root on both LHS and RHS, we get b equivalent to ± 2a
Hence, the end of the latus rectum of a parabola are L = (a, 2a) and L’ (a, -2a)
Thus, the length of the latus rectum of a parabola, L L’’ is 4a.
Length of the Latus Rectum of Hyperbola
Latus rectum of a hyperbola is defined symmetrically as in the case of ellipse and parabola.
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The end of the latus rectum of a parabola is termed (ae, ± b2/ a2) and the length of the latus rectum is termed as 2b2/a.
Latus Rectum Solved Examples
1. What will be the length of the latus rectum whose parabola equation is y2= 12x
Solution:
y2 = 2x
y2 = 4 (3) x
As y2 = 4ax is the equation of a parabola, we get the value of a.
Therefore, the value of a=3
Thus, the length of the latus rectum of a parabola is 4a= 4(3) = 12.
2. What will be the length of the latus rectum of the following parabola x2 = - 4y.
Solution: From the equation given above, we can conclude that the parabola is symmetric about the Y-axis and it is open in a downward position.
x2 = - 4y
x2 = - 4ay
4a = 4
Thus, the length of the latus rectum of a given parabola is 4 units.
3. Calculate the length of the latus rectum of the following parabola x2- 2x + 8y + 17= 0.
Solution: To calculate the value a, we will first convert the above equation in standard form.
x2 - 2x = - 8y -17= 0
x2 - 2x (1) + 12 -12= - 8y -17
(x - 1)2 = - 8y - 17 + 1
(x - 1)2 = - 8y - 16
(x - 1)2 = - 8 (y + 2)
From this equation, we can conclude that the given parabola is symmetric about the y-axis and it is open in an upward position.
Length of the latus rectum = 4a
4a = 8
Thus, the length of the latus rectum of the given parabola is 8 units.
Fun Facts
Conic section was introduced by Menaechmus around 360 - 350 B.C
Euclid gave the name to the parabola around 300 BC.
FAQs on Latus Rectum of Conic Section
1. Explain the conic section.
Circle, Hyperbola, parabola, ellipse are different types of curves in Geometry. These curves are determined as conic sections or conics as we get these curves from the intersection of a plane with a double-napped right circular cone. These curves are used in various fields such as automobiles, headlights, antennas, telescopes, and reflectors.
We get the following situation when the plane cuts the cone other than the vertex.
When β= 90°, the section is determined as a circle
When α < β < 90°, the section is determined as an ellipse
When α = β, the section is determined as a parabola
When 0 ≤ β < α, the section is determined as a hyperbola
Where β is considered as the angle formed by the plane through the vertical axis of the cone.
2. Explain the latus rectum of an ellipse.
In terms of locus, an ellipse is the set of all those points on the XY- plane, whose distance from the two fixed points ( also known as foci) sum up to a constant value.
The line segment perpendicular to the major axis passing through any of the foci such that their endpoint lies on the ellipse is known as the latus rectum of an ellipse.
The length of the latus rectum of an ellipse is 2b2/a
Length = 2b2/a
Here a and b are considered as the length of the minor and major axis.
3. What is the history of the conic section in mathematics?
Conic sections were explored by ancient Greek mathematicians, culminating in Apollonius of Perga's methodical study on their characteristics around 200 BC. Many of the features that identify conic sections in the Euclidean plane may be utilised as alternative definitions.
The three forms of conic sections seem extremely different on the Euclidean plane, although they have many features. The apparent difference vanishes when the Euclidean plane is extended to include a line at infinity, resulting in a projective plane: the branches of a hyperbola meet in two points at infinity, forming a single closed curve; and the two ends of a parabola meet to form a closed curve tangent to the line at infinity.
The ability to see this unification algebraically is provided by further extension, which involves expanding the real coordinates to accommodate complex coordinates.
4. What are the applications of conic sections?
When two heavy objects interact according to Newton's law of universal gravitation, the orbits are conic sections if their shared centre of mass is believed to be at rest. They will both sketch out ellipses if they are tied together; if they are moving apart, they will both follow parabolas or hyperbolas.
Conic sections are utilised in the construction of searchlights, radio telescopes, and some optical telescopes because of their reflecting qualities. A searchlight utilises a parabolic mirror as a reflector, with a bulb at the focus, and a parabolic microphone uses a similar design. The main parabolic mirror reflects light towards a secondary hyperbolic mirror, which reflects it back to a focus behind the first mirror, at the 4.2 metres Herschel optical telescope on La Palma, Canary Islands.