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Latus Rectum of Conic Section‌

iacst-2022

A conic section is defined as the curved surface formed from the intersection of a cone and a plane. In mathematics, three types of conic sections occur, viz. ellipse, parabola, and hyperbola. Many terms are associated with these curves, like directrix, asymptote, latus rectum, focus, etc. Our point of concern will be the latus rectum in this article.

Definition

The word latus rectum is derived from Latin, meaning side, and rectum, meaning straight. Thus, the latus rectum is a chord parallel to the directrix, passing through the focus in a conic section.

Length of a latus rectum of a parabola

From the image above, we need to assume the ends of the latus rectum of the parabola, y2 = 4ax be L and L’. The x-coordinates of L and L’ are equal to ‘a’ as S = (a, 0)

We need to consider the coordinates of L = (a, b).

Since L is a point of the parabola, we will get

b2 = 4a (a) = 4a2

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

Therefore, the ends of the latus rectum of a parabola are L = (a, 2a),

and L’ = (a, -2a) Therefore, the length of the latus rectum of a parabola, LL’ is 4a.

Length of latus rectum of hyperbola

The length of the latus rectum of a hyperbola is given by 2b2/a, where a and b are the arms of a hyperbola.

Latus rectum of conic sections

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

Example 1

What is the length of the latus rectum whose parabola is given by y² = 8x?

Solution

We can write 8 as 2 x 4.

Therefore, we have the equation of parabola as y² = 4(2)x.

Comparing this equation with the standard equation of parabola y² = 4ax, we get, a = 2.

Hence, the length of the latus rectum is 4a = 8.

Example 2

What is the length of the latus rectum of ellipse 4x² + 9y² – 24x + 36y – 72 = 0?

Solution

The given equation of ellipse is 4x² + 9y² – 24x + 36y – 72 = 0

⇒ (4x² – 24x) + (9y² + 36y) – 72 = 0

⇒ 4 (x² - 6x) + 9(y² + 4y) – 72 = 0

⇒ 4 [x² – 6x +9] + 9 [y² + 4y +4] = 144

⇒ 4 (x – 3)² + 9 (y + 2)² = 144

⇒ {(x – 3)²/ 36} + {(y + 2)²/ 16} = 1

⇒ {(x – 3)²/ 6²} + {(y + 2)²/ 4²} = 1

⇒ a = 3 and b = 2

The length of latus rectum of an ellipse is represented by-

= 2b²/a

= 2(2)² /3

= 2(4)/3

= 8/3

Hence, the length of the latus rectum of the ellipse is 8/3.

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