In this section, you will learn how to find vertex, focus, equation of directrix and length of latus rectum of the parabola.
Before seeing example problems, let us remember some basic concepts about parabola.
Parabola symmetric about x-axis and open right ward :
Standard form of parabola
y2 = 4ax
Parabola symmetric about x-axis and open left ward :
Standard form of parabola
y2 = -4ax
Parabola symmetric about y-axis and open up ward :
Standard form of parabola
x2 = 4ay
Parabola symmetric about y-axis and open down ward :
Standard form of parabola
x2 = -4ay
Now let us see some examples based on the above concept.
Example 1 :
Find the focus, vertex, equation of directrix and length of the latus rectum of the parabola
x2 = 5y
Solution :
From the given equation, the parabola is symmetric about y - axis and it is open upward.
x2 = 5y
4a = 5
a = 5/4
Vertex : V (0, 0)
Focus : F (0, 5/4)
Equation of directrix : y = -5/4
Length of latus rectum : 4a = 4(5/4) ==> 5
Example 2 :
Find the focus, vertex, equation of directrix and length of the latus rectum of the parabola
x2 - 8y - 2x + 17 = 0
Solution :
x2 - 8y - 2x + 17 = 0
x2 - 2x = 8y - 17
x2 - 2x + 12 - 12 = 8y - 17
(x - 1)2 = 8y - 17 + 1
(x - 1)2 = 8y - 16
(x - 1)2 = 8(y - 2)
From the given equation, the parabola is symmetric about y - axis and it is open upward.
Let X = x - 1 and Y = y - 2
X2 = 8Y
4a = 8
a = 2
Referred to X and Y X = x - 1 and Y = y - 2 |
Referred to x and y x = X + 1 and y = Y + 2 |
Vertex (0, 0) Focus (0, 2) Equation of directrix Y = -a Y = -2 Length of latus rectum : 4a = 4(2) = 8 |
Vertex (1, 2) Focus (1, 4) Equation of directrix Y = 0 Length of latus rectum : 4a = 4(2) = 8 |
Example 3 :
Find the focus, vertex, equation of directrix and length of the latus rectum of the parabola
x2 = -16y
Solution :
From the given equation, the parabola is symmetric about y - axis and it is open downward.
x2 = -16y
4a = 16
a = 4
Vertex : V (0, 0)
Focus : F (0, -4)
Equation of directrix : y = a ==> y = 4
Length of latus rectum : 4a = 4(4) ==> 16
Example 4 :
Find the focus, vertex, equation of directrix and length of the latus rectum of the parabola
x2 + 4y - 6x + 17 = 0
Solution :
x2 + 4y - 6x + 17 = 0
x2 - 6x = -4y - 17
x2 - 6x + 32 - 32 = -4y - 17
(x - 3)2 = -4y - 17 + 9
(x - 3)2 = -4y - 8
(x - 3)2 = -4(y + 2)
From the given equation, the parabola is symmetric about y - axis and it is open downward.
Let X = x - 3 and Y = y + 2
X2 = -4Y
4a = 4
a = 1
Referred to X and Y X = x - 3 and Y = y + 2 |
Referred to x and y x = X + 3 and y = Y - 2 |
Vertex (0, 0) Focus (0, -1) Equation of directrix Y = a Y = 1 Length of latus rectum : 4a = 4(1) = 4 |
Vertex (3, -2) Focus (3, -3) Equation of directrix Y = -1 Length of latus rectum : = 4 |
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