Posts Tagged ‘directrix’

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 1082

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

Focus (6,0), directrix x = -6

 

Solution:-

 

First, notice that since the directrix is a vertical line, the parabola will open either to the left or to the right, not up or down. Since the directrix is to the left of the focus, and a parabola always opens away from its directrix, this parabola will open to the right.

The vertex will be halfway between the focus and the directrix. It will have the same y-coordinate as the y-coordinate of the focus.

The vertex of the parabola is (\frac{6+(-6)}{2},0), which simplifies to (0,0).

The parabola with directrix x = -p and vertex (0,0) will have the equation y2 = 4px.

Since the directrix of this parabola is x = -6 , p = 6.

The equation of the parabola with directrix x = -6 and focus (6,0) is  y2 = 4*6x, which simplify to y2 = 24x.