Posts Tagged ‘parabola’

Problem 1083

Find the equation if the parabola determined by the given information.

Focus (3,5), directrix x = -1.

 

Solution:-

 

Notice that the directrix is a vertical line. Since the axis of symmetry is perpendicular to the directrix, then it is a horizontal line.

The parabola with a horizontal axis of symmetry will have directrix x = h – p, focus (h +p, k), and standard equation (y – k)2 = 4p(x  – h).

Since the directrix of this parabola is x = -1, then – 1 = h – p.

Since the x – value of the focus is 3, then 3 = h + p.

Find h and p using any method for solving a system of equations. Using the elimination method, eliminate p and solve for h.

h = 1

Substitute the value for h into one of the equation to solve for p.

p = 2

k is the y-value of the focus. So, k = 5.

Substitute the values for h, p, and k into the standard equation of a parabola.

(y – 5)2 = 4 *2(x – 1)

(y – 5)2 = 8(x – 1)

 

 

Problem 717

Find the vertex of the parabola x = 3y2 + 6y+1

 

Solution:-

 

y coordinate =\frac{ -b}{2a}

= \frac{-6}{(2*3)}

= -1
x coordinate = 3(-1)^{2}+6(-1)+1

=3-6+1

=-2
So vertex of the parabola is (-2,-1)

 

Find the equation of the parabola determined by the given information.

Find the equation of the parabola determined by the given information.

Focus (3,5), directrix x = -1

 

Solution:-

Notice that the directrix is a vertical line. Since the axis of symmetry is perpendicular to the directrix, then it is a horizontal line.

The parabola with a horizontal axis of symmetry will have directrix x = h –p,

focus (h + p,k), and standard equation (y - k)^{2} = 4p(x – h).

Since the directrix of the parabola is x = -1, then -1, then -1 = h – p.

Since the x-value of the focus is 3, then 3 = h + p.

Find h and  p using any method for solving a  system of equation. Using the elimination method, eliminate p and solve for h.

h  = 1

substitute the value for h into one of the equation to solve for p.

p = 2

k is the y – value for h , p, and k into standard equation of a parabola.

(y - 5)^{2}= 4*2(x – 1)

(y - 5)^{2}= 8(x – 1)