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Parabola – vertex, focus, directrix, latus rectum
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Parabola – vertex, focus, directrix, latus rectum

by Namagwa Benard
(Kenya)











































Sideways, or Horizontal, Parabola

   • Given

           4y² - 8y + 3x - 2 = 0

   • Show

           the equation represents a parabola

   • Find the:

           vertex

           focus

           directrix

           length of latus rectum

Comments for Parabola – vertex, focus, directrix, latus rectum

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Oct 14, 2012
Sideways Parabola
by: Staff


Answer:


Part I

4y² - 8y + 3x - 2 = 0 represents a sideways, or horizontal, parabola.

         Solve for x

4y² - 8y + 3x - 2 = 0  

Subtract 4y² from both sides of the equation


4y² - 8y + 3x - 2 - 4y² = 0 - 4y²

4y² - 4y² - 8y + 3x - 2 = 0 - 4y²

0 - 8y + 3x - 2 = 0 - 4y²

- 8y + 3x - 2 = - 4y²

Add 8y to both sides of the equation

- 8y + 3x - 2 + 8y= - 4y² + 8y

- 8y + 8y + 3x - 2 = - 4y² + 8y

0 + 3x - 2 = - 4y² + 8y

3x - 2 = - 4y² + 8y

Add 2 to both sides of the equation

3x - 2 + 2 = - 4y² + 8y + 2

3x + 0 = - 4y² + 8y + 2

3x = - 4y² + 8y + 2

Divide each side of the equation by 3

3x / 3 = (- 4y² + 8y + 2) / 3

x * (3 / 3) = (- 4y² + 8y + 2) / 3

x * (1) = (- 4y² + 8y + 2) / 3

x = (- 4y² + 8y + 2) / 3

x = (-4/3)y² + (8/3)y + 2/3


         Plot the equation


 Parabola (sideways, or horizontal, parabola): 4y² - 8y + 3x - 2 = 0

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Oct 14, 2012
Sideways Parabola
by: Staff

-------------------------------------------------

Part II

The vertex of the parabola is (2,1)

         You can solve for the vertex of the parabola using the first term of the quadratic equation.
the quadratic formula is:


x = (-b/2a) ± [√(b² - 4ac)/2a]

Because this is a sideways parabola, the x and y variables must be reversed.

y = (-b/2a) ± [√(b² - 4ac)/2a]

the x coordinate of the vertex of the parabola is:

y_vertex = (-b/2a)

for your equation:

x = (-4/3)y² + (8/3)y + 2/3

a = (-4/3)

b = (8/3)

y_vertex = -(8/3) / [2*(-4/3)]

y_vertex = -(8/3) / (-8/3)

y_vertex = 1

now that you know the y coordinate of the vertex, substitute this value in the original equation to compute x

x = (-4/3)y² + (8/3)y + 2/3

x_vertex = (-4/3)*1² + (8/3)*1 + 2/3

x_vertex = (-4/3)*1 + (8/3)*1 + 2/3

x_vertex = (-4/3) + (8/3) + 2/3

x_vertex = (-4 + 8 + 2)/3

x_vertex = (6)/3

x_vertex = 2

the coordinates of the vertex in x,y format are: (2, 1)



         You can also compute the vertex by rewriting the equation in vertex format as follows:
 

x = (-4/3)y² + (8/3)y + 2/3

x = [(-4/3)y² + (8/3)y] + 2/3

x = [(-4/3)y² + 2*(4/3)y] + 2/3

x = (-4/3)[y² - 2*1y ] + 2/3

x = (-4/3)[y² - 2*1y + 1 - 1 ] + 2/3

x = (-4/3)[y² - 2*1y + 1] + 4/3 + 2/3

x = (-4/3)(y - 1)² + 4/3 + 2/3

x = (-4/3)(y - 1)² + (4 + 2)/3

x = (-4/3)(y - 1)² + (6)/3

x = (-4/3)(y - 1)² + 2

As you can see, the coordinates of the vertex in x,y format are: (2, 1)



The focus of the parabola is (29/16, 1)


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Oct 14, 2012
Sideways Parabola
by: Staff


-------------------------------------------------

Part III


The Standard equation of a parabola with a horizontal axis is:


(y - k)² = 4p(x - h)

The coordinates of the vertex is (h,k)

Substituting the values for the vertex (2, 1) for your equation:

(y - 1)² = 4p(x - 2)

To solve for p, enter in a point on the curve, such as (2/3, 0).

when y = 0, x = 2/3

x = (-4/3)y² + (8/3)y + 2/3

x = (-4/3)*0² + (8/3)*0 + 2/3

x = 0 + 0 + 2/3

x = 2/3


(0 - 1)² = 4p(2/3 - 2)

(- 1)² = 4p(2/3 - 6/3)

1 = 4p(- 4/3)

1 = p(- 16/3)

1 * (-3/16) = p(- 16/3) * (-3/16)

-3/16 = p * 1


-3/16 = p

p = -3/16


the final equation is:


(y - 1)² = 4 * -3/16 * (x - 2)

(y - 1)² = (-3/4) * (x - 2)


Since p = -3/16, the focus is 3/16 units to the left of the vertex.

The focus = (2 - 3/16, 1)

= (32/16 - 3/16, 1)

= (29/16, 1)

the coordinates of the focus in x,y format are: (29/16, 1)



The directrix of the parabola is x = 35/16
Since p = -3/16, the directrix is 3/16 units to the right of the vertex.


The directrix = 2 + 3/16

= 32/16 + 3/16

= 35/16

the x coordinate of the directrix is: 35/16



The length of the latus rectum of the parabola is 3/4

length of latus rectum =  4|p| 


length of latus rectum = 4|-3/16| = 12/16 = 3/4


the length of latus rectum is 3/4




Thanks for writing.

Staff
www.solving-math-problems.com


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