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:
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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.
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Conic Section |
Length of the Latus Rectum |
End of the Latus Rectum |
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Y2 = 4ax |
4a |
L= ( a, 2a) , L’ (a, -2a) |
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(x2 / a2) + (y2 /b2) = 1 |
If a > b, 2b2 / a |
L = (ae, b2 a) L’ (ae- b2 a) |
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(x2 / a2) + (y2 /b2) = 1 |
If b > a, 2a2 / b |
L = (ae, b2 a) L’ (ae- b2 a) |
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(x2 / a2) – (y2 /b2) = 1 |
2b2/ a |
L = (ae, b2 a) L’ (ae- b2 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.