Writing The Equation Of A Parabola
PARABOLA FORMULAS
Parabola Opens Right
Standard equation of a parabola that opens right and symmetric about x-axis with vertex at origin.
y2 = 4ax
Standard equation of a parabola that opens up and symmetric about x-axis with at vertex (h, k).
(y - k)2 = 4a(x - h)
Graph of y2 = 4ax :
Axis of symmetry : x -axis
Equation of axis : y = 0
Vertex : V(0, 0)
Focus : F(a, 0)
Equation of latus rectum : x = a
Equation of directrix : x = -a
Length of latus rectum : 4a
Distance between the vertex and focus = a.
Distance between the directrix and vertex = a.
Distance between directrix and latus rectum = 2a.
Parabola Opens Left
Standard equation of a parabola that opens left and symmetric about x-axis with vertex at origin.
y2 = -4ax
Standard equation of a parabola that opens up and symmetric about x-axis with at vertex (h, k).
(y - k)2 = -4a(x - h)
Graph of y2 = -4ax :
Axis of symmetry : x -axis
Equation of axis : y = 0
Vertex : V(0, 0)
Focus : F(-a, 0)
Equation of latus rectum : x = -a
Equation of directrix : x = a
Length of latus rectum : 4a
Distance between the vertex and focus = a.
Distance between the directrix and vertex = a.
Distance between directrix and latus rectum = 2a.
Parabola Opens Up
Standard equation of a parabola that opens up and symmetric about y-axis with vertex at origin.
x2 = 4ay
Standard equation of a parabola that opens up and symmetric about y-axis with at vertex (h, k).
(x - h)2 = 4a(y - k)
Graph of x2 = 4ay :
Axis of symmetry : y-axis
Equation of axis : x = 0
Vertex : V(0, 0)
Focus : F(0, a)
Equation of latus rectum : y = a
Equation of directrix : y = -a
Length of latus rectum : 4a
Distance between the vertex and focus = a.
Distance between the directrix and vertex = a.
Distance between directrix and latus rectum = 2a.
Parabola Opens Down
Standard equation of a parabola that opens up and symmetric about y-axis with vertex at origin.
x2 = -4ay
Standard equation of a parabola that opens up and symmetric about y-axis with at vertex (h, k).
(x - h)2 = -4a(y - k)
Graph of x2 = -4ay :
Axis of symmetry : y-axis
Equation of axis : x = 0
Vertex : V(0, 0)
Focus : F(0, -a)
Equation of latus rectum : y = -a
Equation of directrix : y = a
Length of latus rectum : 4a
Distance between the vertex and focus = a.
Distance between the directrix and vertex = a.
Distance between directrix and latus rectum = 2a.
Solved Problems
Problem 1 :
Find the vertex, focus, directrix, latus rectum of the following parabola :
x2 = -16y
Solution :
x 2 = -16y is in the form of x 2 = -4ay.
So, the given parabola opens down and symmetric about y-axis with vertex at (0, 0).
Comparing x 2 = -16y andx 2 = -4ay,
4a = 16
Divide each side by 4.
a = 4
Focus : F(0, -a) = F(0, -4).
Equation of latus rectum : y = -a ----> y = -4.
Equation of directrix : y = a ----> y = 4.
Problem 2 :
Find the vertex, focus, directrix, latus rectum of the following parabola :
y2 - 8y - x + 19 = 0
Solution :
Write the equation of parabola in standard form.
y2 - 8y = x - 19
y2 - 2(y)(4) + 42 - 42 = x - 19
(y - 4)2 - 42 = x - 19
(y - 4)2 - 16 = x - 19
Add 16 to each side.
(y - 4)2 = (x - 3)
(y - 4)2 = (x - 3) is in the form of (y - k) 2 = 4a(x - h).
So, the parabola opens up and symmetric about x-axis with vertex at (h, k) = (3, 4).
Comparing (y - 4)2 = (x - 3) and (y - k)2 = 4a(x - h),
4a = 1
Divide each side by 4.
a = 1/4 = 0.25
Standard form equation of the given parabola :
(y - 4) 2 = (x - 3)
Let Y = y - 4 and X = x - 3.
Then,
Y2 = X
Referred to X and Y
Referred to x and y
Vertex
(0, 0)
X = 0, Y = 0
x - 3 = 0, y - 4 = 0
x = 3, y = 4
(3, 4)
Focus
X = 0.25, Y = 0
x - 3 = 0.25, y - 4 = 0
x = 3.25, y = 4
(3.25, 4)
Latus rectum
X = 0.25
x - 3 = 0.25
x = 3.25
Directrix
X = -0.25
x - 3 = -0.25
x = 2.75
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Writing The Equation Of A Parabola
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