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Math  /  Geometry

QuestionFind the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation.
Directrix the line y=17y=\frac{1}{7}; vertex at (0,0)(0,0)
What is the equation of the parabola? □\square (Use integers or fractions for any numbers in the equation.)

Studdy Solution

STEP 1

1. The parabola has a vertex at the origin (0,0)(0,0).
2. The directrix is a horizontal line given by y=17 y = \frac{1}{7} .
3. The parabola opens vertically since the directrix is horizontal.

STEP 2

1. Determine the orientation and focus of the parabola.
2. Use the vertex form of a parabola to write the equation.
3. Find the length of the latus rectum and its endpoints.
4. Graph the parabola and the latus rectum.

STEP 3

Determine the orientation and focus of the parabola:
Since the directrix is y=17 y = \frac{1}{7} and the vertex is at (0,0)(0,0), the parabola opens downwards. The distance from the vertex to the directrix is 17\frac{1}{7}, so the focus is at (0,−17)(0, -\frac{1}{7}).

STEP 4

Use the vertex form of a parabola to write the equation:
The standard form of a parabola that opens vertically is (x−h)2=4p(y−k) (x-h)^2 = 4p(y-k) , where (h,k)(h,k) is the vertex and pp is the distance from the vertex to the focus. Here, h=0h = 0, k=0k = 0, and p=−114p = -\frac{1}{14}.
Thus, the equation is:
x2=4(−114)y x^2 = 4\left(-\frac{1}{14}\right)y x2=−27y x^2 = -\frac{2}{7}y

STEP 5

Find the length of the latus rectum and its endpoints:
The length of the latus rectum is ∣4p∣|4p|, which is ∣4×−114∣=414=27\left|4 \times -\frac{1}{14}\right| = \frac{4}{14} = \frac{2}{7}.
The endpoints of the latus rectum are horizontally centered at the focus (0,−17)(0, -\frac{1}{7}) and extend 17\frac{1}{7} units to the left and right:
Endpoints: (−17,−17)\left(-\frac{1}{7}, -\frac{1}{7}\right) and (17,−17)\left(\frac{1}{7}, -\frac{1}{7}\right).

STEP 6

Graph the parabola and the latus rectum:
1. Plot the vertex at (0,0)(0,0).
2. Plot the focus at (0,−17)(0, -\frac{1}{7}).
3. Draw the directrix as the line y=17y = \frac{1}{7}.
4. Plot the endpoints of the latus rectum at (−17,−17)\left(-\frac{1}{7}, -\frac{1}{7}\right) and (17,−17)\left(\frac{1}{7}, -\frac{1}{7}\right).
5. Sketch the parabola opening downwards through these points.
The equation of the parabola is:
x2=−27y \boxed{x^2 = -\frac{2}{7}y}

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