Posts Tagged ‘Focus’

Problem 1084

Find the vertex, focus, and directrix of the following parabola. Then draw the graph.

(x – 2)2 = -3 (y + 1)

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Solution:-

 

The two standard forms of the equation for a parabola are (x – h)2 = 4p(y – k) or

(y – k)2 = 4p(x – k).

Since the parabola is already in one of the standard forms, you can determine the values of h, k, and p, and form these values determine the vertex, focus, and directrix.

h = 2, k = -1, and p = –\frac{3}{4}

Remember, for a parabola written in the form (x – h)2 = 4p(y – k), the vertex is (h , k).

Thus , the vertex of this parabola is (2,-1).

The axis of symmetry is vertical. The focus is –\frac{3}{4} units away from the vertex, in the direction of the axis of symmetry.

Thus, it will be \frac{3}{4} units below the vertex.

The focus is (2,-\frac{7}{4}).

The directrix will be \frac{3}{4} units above the vertex.

The equation for the directrix is

y = –\frac{1}{4}

With this information, graph the parabola. First, graph the vertex, focus, and axis of symmetry.

Next, fill in the graph of the 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 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.