Parabola
See how to solve a parabola
(focus, directrix, equation, latus rectum).
8 examples and their solutions.
Definition
(point ~ focus) = (point ~ directrix).
Focus: blue point
Directrix: dashed line
Parabola: y2 = 4px
Equation
y2 = 4px
Focus: (p, 0)
Directrix: x = -p
Focus: (p, 0)
Directrix: x = -p
Example
y2 = 8x
1. Focus?
2. Equation of the directrix?
Solution 1. Focus?
2. Equation of the directrix?
y2 = 8x
= 4⋅2⋅x
1. (2, 0)
2. x = -2
= 4⋅2⋅x
1. (2, 0)
2. x = -2
Example
Focus: (3, 0)
Directrix: x = -3
Equation of the parabola?
Solution Directrix: x = -3
Equation of the parabola?
Focus: (3, 0)
Directrix: x = -3
y2 = 4⋅3⋅x
y2 = 12x
Directrix: x = -3
y2 = 4⋅3⋅x
y2 = 12x
Example
Focus: (3, 4)
Directrix: x = -1
Equation of the parabola?
Solution Directrix: x = -1
Equation of the parabola?
2p = 3 + (-1)
2p = 4
p = 2
(2, 0) → (3, 4) = (2 + 1, 0 + 4)
(x, y) → (x + 1, y + 4) - [2]
(y - 4)2 = 4⋅2⋅(x - 1) - [3]
(y - 4)2 = 8(x - 1)
[1]
Draw the parabola, the focus, and the directrix.
The vertex of the parabola
is the midpoint of the focus and the directrix.
So set
(focus ~ vertex) = (vertex ~ dirextrix) = p.
The vertex of the parabola
is the midpoint of the focus and the directrix.
So set
(focus ~ vertex) = (vertex ~ dirextrix) = p.
[2]
p = 2
So the focus should be (2, 0).
But the focus is (3, 4).
Then there's a translation
(2, 0) → (3, 4).
(3, 4) = (2 + 1, 0 + 4)
So the translation is
(x, y) → (x + 1, y + 4).
So the focus should be (2, 0).
But the focus is (3, 4).
Then there's a translation
(2, 0) → (3, 4).
(3, 4) = (2 + 1, 0 + 4)
So the translation is
(x, y) → (x + 1, y + 4).
[3]
p = 2
(x, y) → (x + 1, y + 4)
Then the equation of the parabola is
(y - 4)2 = 4⋅2⋅(x - 1).
(x, y) → (x + 1, y + 4)
Then the equation of the parabola is
(y - 4)2 = 4⋅2⋅(x - 1).
Close
Latus Rectum
y2 = 4px
(latus rectum) = |4p|
(latus rectum) = |4p|
Example
y2 = 8x
Latus rectum?
Solution Latus rectum?
(latus rectum) = |8|
= 8
= 8
Example
y2 = -12x
Latus rectum?
Solution Latus rectum?
(latus rectum) = |-12|
= 12
= 12
Parabola: x2 = 4py
Equation
x2 = 4py
Focus: (0, p)
Directrix: y = -p
Focus: (0, p)
Directrix: y = -p
Example
y = x2
1. Focus?
2. Equation of the directrix?
Solution 1. Focus?
2. Equation of the directrix?
y = x2
x2 = 4
= 4⋅14⋅y
1. (0, 14)
2. y = -14
x2 = 4
= 4⋅14⋅y
1. (0, 14)
2. y = -14
Close
Example
Focus: (0, 2)
Directrix: y = -2
Equation of the parabola?
Solution Directrix: y = -2
Equation of the parabola?
Focus: (0, 2)
Directrix: y = -2
x2 = 4⋅2⋅y
x2 = 8y
Directrix: y = -2
x2 = 4⋅2⋅y
x2 = 8y
Latus Rectum
x2 = 4py
(latus rectum) = |4p|
(latus rectum) = |4p|
Example
y = 3x2
Latus rectum?
Solution Latus rectum?
y = 3x2
3x2 = y
x2 = 13y
(latus rectum) = |13|
= 13
3x2 = y
x2 = 13y
(latus rectum) = |13|
= 13
Close