Standard Equation of a Parabola
Standard form of a parabola is y = ax2 + bx + c where a, b, and c are real numbers and a is not equal to zero. A parabola is defined as the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane.
In this article, we will understand what is a Parabola, the standard equation of a Parabola, related examples and others in detail.
Table of Content
- What is a Parabola?
- Equation of a Parabola
- General Equations of a Parabola
- Standard Equations of a Parabola
- Parts of a Parabola
- Examples on Equation of a Parabola
What is a Parabola?
A parabola is a conic section defined as the set of all points equidistant from a point called the focus and a line called the directrix. The standard equations for a parabola depend on its orientation (opening direction) and position.
Equation of a Parabola
Equation of parabola can be written in standard form or general form and both of them are added below:
General Equations of a Parabola
The general equation of a parabola is,
y = 4a(x β h)2 + k
(or)
x = 4a(y β k)2 + h
Where (h, k) is the vertex of a parabola.
Standard Equations of a Parabola
The standard equation of a parabola is,
y = ax2 + bx + c
(or)
x = ay2 + by + c
where, a can never be zero.
Parts of a Parabola
Some important terms and parts of a parabola are:
- Focus: Focus is the fixed point of a parabola.
- Directrix: The directrix of a parabola is the line perpendicular to the axis of a parabola.
- Focal Chord: The chord that passes through the focus of a parabola, cutting the parabola at two distinct points, is called the focal chord.
- Focal Distance: The focal distance is the distance of a point (x1, y1) on the parabola from the focus.
- Latus Rectum: A latus rectum is a focal chord that passes through the focus of a parabola and is perpendicular to the axis of the parabola. The length of the latus rectum is LLβ = 4a.
- Eccentricity: The ratio of the distance of a point from the focus to its distance from the directrix is called eccentricity (e). For a parabola, eccentricity is equal to 1, i.e., e = 1.
A parabola has four standard equations based on the orientation of the parabola and its axis. Each parabola has a different transverse axis and conjugated axis.
Equation of Parabola |
Parabola |
Formulae of Parameters of a Parabola |
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y2 = 4ax |
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y2 = -4ax |
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x2 = 4ay |
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x2 = -4ay |
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The following are the observations made from the standard form of equations of a parabola:
- A parabola is symmetrical w.r.t its axis. For example, y2 = 4ax is symmetric w.r.t the x-axis, whereas x2 = 4ay is symmetric with respect to the y-axis.
- If a parabola is symmetric about the x-axis, then the parabola opens towards the right if the x-coefficient is positive and towards the left if the x-coefficient is negative.
- If a parabola is symmetric about the y-axis, then the parabola opens upwards if the y-coefficient is positive and downwards if the y-coefficient is negative.
The following are the standard equations of a parabola when the axis of symmetry is either parallel to the x-axis or y-axis and the vertex is not at the origin.
Equation of Parabola |
Parabola |
Formulae of Parameters of a Parabola |
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(y β k)2 = 4a(x β h) |
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(y β k)2 = -4a(x β h) |
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(x β h)2 = 4a(y β k) |
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(x β h)2 = -4a(y β k) |
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Equation of Parabola Derivation
Let P be a point on the parabola whose coordinates are (x, y). From the definition of a parabola, the distance of point P to the focus (F) is equal to the distance of the same point P to the directrix of a parabola. Now, let us consider a point X on the directrix, whose coordinates are (-a, y).
From the definition of the eccentricity of a parabola, we have
e = PF/PX = 1
β PF = PX
The coordinates of the focus are (a, 0). Now, by using the coordinate distance formula, we can find the distance of point P (x, y) to the focus F (a, 0).
PF = β[(x β a)2 + (y β 0)2]
β PF = β[(x β a)2 + y2] ββββββ (1)
The equation of the directrix is x + a = 0. To find the distance of PX, we use the perpendicular distance formula.
PX = (x + a)/β[12 + 02]
β PX = x +a ββββββ (2)
We already know that PF = PX. So, equate equations (1) and (2).
β[(x β a)2 + y2] = (x + a)
By, squaring on the both sides we get,
β [(x β a)2 + y2] = (x + a)2
β x2 + a2 β 2ax + y2 = x2 + a2 + 2ax
β y2 β 2ax = 2ax
β y2 = 2ax + 2ax β y2 = 4ax
Thus, we have derived the equation of a parabola. Similarly, we can derive the standard equations of the other three parabolas.
- y2 = -4ax
- x2 = 4ay
- x2 = -4ay
y2 = 4ax, y2 = -4ax, x2 = 4ay, and x2 = -4ay are the standard equations of a parabola.
Articles Related to Parabola:
Examples on Equation of a Parabola
Example1: Find the length of the latus rectum, focus, and vertex, if the equation of the parabola is y2 = 12x.
Solution:
Given,
Equation of the parabola is y2 = 12x
By comparing the given equation with the standard form y2 = 4ax
4a = 12
β a = 12/4 = 3
We know that,
Latus rectum of a parabola = 4a = 4 (3) = 12
Now, focus of the parabola = (a, 0) = (3, 0)
Vertex of the given parabola = (0, 0)
Example 2: Find the equation of the parabola which is symmetric about the X-axis, and passes through the point (-4, 5).
Solution:
Given,
Parabola is symmetric about the X-axis and has its vertex at the origin.
Thus, the equation can be of the form y2 = 4ax or y2 = -4ax, where the sign depends on whether the parabola opens towards the left side or right side.
Parabola must open left since it passes through (-4, 5) which lies in the second quadrant.
So, the equation will be: y2 = -4ax
Substituting (-4, 5) in the above equation,
β (5)2 = -4a(-4)
β 25 = 16a
β a = 25/16
Therefore, the equation of the parabola is: y2 = -4(25/16)x (or) 4y2 = -25x.
Example 3: Find the coordinates of the focus, the axis, the equation of the directrix, and the latus rectum of the parabola x2 = 16y.
Solution:
Given,
Equation of the parabola is: x2 = 16y
By comparing the given equation with the standard form x2 = 4ay,
4a = 16 β a = 4
Coefficient of y is positive so the parabola opens upwards.
Also, the axis of symmetry is along the positive Y-axis.
Hence,
Focus of the parabola is (a, 0) = (4, 0).
Equation of the directrix is y = -a, i.e. y = -4 or y + 4 = 0.
Length of the latus rectum = 4a = 4(4) = 16.
Example 4: Find the length of the latus rectum, focus, and vertex if the equation of a parabola is 2(x-2)2 + 16 = y.
Solution:
Given,
Equation of a parabola is 2(x-2)2 + 16 = y
By comparing the given equation with the general equation of a parabola y = a(x β h)2 + k, we get
a = 2
(h, k) = (2, 16)
We know that,
Length of latus rectum of a parabola = 4a
= 4(2) = 8
Now, focus= (a, 0) = (2, 0)
Now, Vertex = (2, 16)
Example 5: Equation of a parabola is x2 β 12x + 4y β 24 = 0, then find its vertex, focus, and directrix.
Solution:
Given,
Equation of the parabola is x2 β 12x + 4y β 24 = 0
β x2 β 12x + 36 β 36 + 4y β 24 = 0
β (x β 6)2 + 4y β 60 = 0
β (x β 6)2 = -4(y + 15)
Obtained equation is in the form of (x β h)2 = -4a(y β k)
-4a = -4 β a = 1
So, the vertex = (h, k) = (6, β 15)
Focus = (h, k β a) = (6, -15-1) = (6, -16)
Equation of the directrix is y = k + a
β y = -15 + 1 β y = -14
β y + 14 = 0
FAQs on Equation of Parabola
How Do you Find the Standard Equation of a Parabola?
Standard form of parabola is y2 = 4ax or x2 = 4ay.
What is the Normal Equation of Parabola?
Equation of normal to the parabola y2 = 4ax with a slope m is given as: y = mx β 2am β am3
How Do You Find the Vertex of a Parabola?
For given parabola: y = ax2 + bx + c its vertex can be found using the formula x = β b/2a. Plug this x value back into the equation to find the corresponding y-coordinate.
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