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Latus rectum of a parabola is the line passing through its foci which is parallel to the directrix of the parabola and perpendicular to the axis of the parabola. A parabola has only one latus rectum whereas an ellipse and a hyperbola have 2 latus rectums.
| Length of Latus Rectum of a Parabola LL’ = 4a |
The focus of the parabola lies exactly at the midpoint of the length of the latus rectum and they are all collinear in nature. For a parabola, the length of the Latus Rectum is 4 times the distance between the focus and the vertex.
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Key Terms : Parabola, Latus Rectum, Derivation of Length, vertex, Hyperbola
What is a Parabola?
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Parabola is the locus of a point that moves in a plane, such that its distance from a fixed point is equal to its perpendicular distance from a fixed straight line. There are four standard equations of a parabola as follows:
- y2 = 4ax
- y2 = – 4ax
- x2 = 4ay
- x2 = – 4ay
Parabola
The important formulas relating to the Latus Rectum of a parabola are tabulated below.
| Parabola | Vertex | Focus | Axis of Symmetry | Directrix | Length of Latus Rectum | Endpoints of Latus Rectum |
|---|---|---|---|---|---|---|
| y2 = 4ax | (0,0) | (a,0) | y = 0 | x = -a | 4a | (a,±2a) |
| y2 = – 4ax | (0,0) | (-a,0) | y = 0 | x = a | 4a | (-a,±2a) |
| x2 = 4ay | (0,0) | (0,a) | x = 0 | y = -a | 4a | (±2a,a) |
| x2 = – 4ay | (0,0) | (0,-a) | x = 0 | y = a | 4a | (±2a,-a) |
Discover about the Chapter video:
Conic Sections Detailed Video Explanation:
Latus Rectum of a Parabola
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Latus rectum of a conic section is a chord that is parallel to the directrix and passes through the focus. The half-length of the Latus Rectum is called Semi-Latus Rectum. The Latus Rectum of a parabola is a line segment perpendicular to the axis of the parabola, which passes through the focus and whose endpoints lie on the parabola.
The length of the latest rectum is denoted by 4a
where a is the distance between the vertex and the focus of the parabola.
The half-length of the Latus Rectum is called Semi-Latus Rectum and it is denoted by 2a.
The endpoints of the Latus Rectum lie on the parabola, which is denoted as L(a,2a) and L’(a,-2a).
Latus Rectum of a Parabola
Two parabolas are said to be equal if they have the Latus Rectum of the same length.
Also Read: Latus Rectum of Ellipse
Derivation of Length of the Latus Rectum
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We know that the endpoints on the Latus Rectum are L(a,2a) and L’(a,-2a). Hence, to find the length of the Latus Rectum, all we have to do is find the distance between the points L and L’.
Using Distance Formula, the length LL’ is
\(\to\) \(\sqrt{[(a - a)² + {2a - (-2a)}²]} \)
\(\to\)[0 + {2a + 2a}2]
\(\to\)[4a2]
\(\to ±\)4a
As the distance cannot be negative, we get the length of the Latus Rectum as 4a.
Things to Remember
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- Latus rectum of a parabola is the line passing through its foci which is parallel to the directrix of the parabola.
- Length of Latus Rectum of a Parabola LL’ = 4a.
- For a parabola, the length of the Latus Rectum is 4 times the distance between the focus and the vertex.
- The end points of the latus rectum of a parabola with standard equation y² = 4ax is (a,±2a)
Sample Questions
Ques. Find the equation of the parabola with vertex at (0, 0) and focus at (0, 2). [2 marks]
Ans: Since the vertex is at (0,0) and the focus is at (0,2) which lies on the y-axis, the y-axis is the axis of the parabola. Therefore, the equation of the parabola is of the form x² = 4ay. Thus, we have,
x2 = 4(2)y
i.e., x2 = 8y
Ques. Find the length of the latus rectum of the parabola y2 - 2y = -2x - 3. [2 marks]
Ans: y2 - 2y = -2x - 3
(y - 1)2 - 1 = -2x - 3
(y - 1)2 = -2x - 3 + 1
(y - 1)2 = -2x - 2
(y - 1)2 = -2(x + 1)
Therefore, the length of the Latus Rectum is 2.
Ques. For parabola y2=84x, what are the endpoints and length of the Latus Rectum? [2 marks]
Ans: The standard equation of the parabola is
y2=4ax
Comparing it with y2=84x, we get
4ax = 84x
4a = 84
a = 21
We know that the endpoints on the Latus Rectum are L(a,2a) and L’(a,-2a).
Therefore, the endpoints are L(21,42) and L’(21,-42).
Also, the length of the Latus Rectum is 84.
Ques. For parabola y²=96x, what are the endpoints and length of the Latus Rectum? [2 marks]
Ans: The standard equation of the parabola is
y2=4ax
Comparing it with y2=96x, we get
4ax = 96x
4a = 96
a = 24
We know that the endpoints on the Latus Rectum are L(a,2a) and L’(a,-2a).
Therefore, the endpoints are L(24,48) and L’(24,-48).
Also, the length of the Latus Rectum is 96.
Ques. For parabola y2=-24x, what are the endpoints and length of the Latus Rectum? [2 marks]
Ans: The standard equation of the parabola is
y2=-4ax
Comparing it with y2=-24x, we get
4ax = 24x
4a = 24
a = 6
We know that the endpoints on the Latus Rectum are L(-a,2a) and L’(-a,-2a).
Therefore, the endpoints are L(-6,12) and L’(-6,-12).
Also, the length of the Latus Rectum is 24.
Ques. For parabola x2= – 36y, what are the endpoints and length of the Latus Rectum? [2 marks]
Ans: The standard equation of the parabola is
x2=-4ay
Comparing it with x2=-36y, we get
4ay = 36y
4a = 36
a = 9
We know that the endpoints on the Latus Rectum are L(2a, -a) and L’(-2a, -a).
Therefore, the endpoints are L(18,-9) and L’(-18,-9).
Also, the length of the Latus Rectum is 36.
Ques. For parabola x2=-72y, what are the endpoints and length of the Latus Rectum? [2 marks]
Ans: The standard equation of the parabola is
x2=-4ay
Comparing it with x2=-72y, we get
4ay = 72y
4a = 72
a = 18
We know that the endpoints on the Latus Rectum are L(2a, -a) and L’(-2a, -a).
Therefore, the endpoints are L(36,-18) and L’(-36,-18).
Also, the length of the Latus Rectum is 72.
Ques. For parabola x2=64y, what are the endpoints and length of the Latus Rectum? [2 marks]
Ans: The standard equation of the parabola is
x2=4ay
Comparing it with x²=64y, we get
4ay = 64y
4a = 64
a = 16
We know that the endpoints on the Latus Rectum are L(2a, a) and L’(-2a, a).
Therefore, the endpoints are L(32,16) and L’(-32,16).
Also, the length of the Latus Rectum is 64.
Also Read:
| Circle | Standard Equation of Circle | Ellipse |
| Eccentricity of Ellipse | Hyperbola | Latus Rectum of Hyperbola |
| Distance between Two Points | Horizontal and Vertical Lines | Angle Between Two Lines |
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