Ask a New Question
Menu

Write the equation of the parabola with a focus at (10,−3) and a directrix of x=2 . It will be helpful to draw a sketch first and identify the information needed to write the equation.(1 point)

Responses

4(x−10)=(y+3)2
4 left parenthesis x minus 10 right parenthesis equals left parenthesis y plus 3 right parenthesis squared

16(x−6)=(y+3)2
16 left parenthesis x minus 6 right parenthesis equals left parenthesis y plus 3 right parenthesis squared

(x−6)2=16(y+3)
left parenthesis x minus 6 right parenthesis squared equals 16 left parenthesis y plus 3 right parenthesis

4(x−6)=(y+3)2

1 answer

To find the equation of a parabola given a focus and a directrix, we start with the definitions:

  • The focus is the point from which distances to points on the parabola are measured.
  • The directrix is a line from which distances to points on the parabola are also measured.
  • The parabola is defined as the set of all points that are equidistant from the focus and the directrix.

Given:

  • The focus is at \( (10, -3) \)
  • The directrix is the line \( x = 2 \)

Steps to Derive the Equation:

  1. Determine the orientation: Since the directrix is a vertical line, the parabola opens horizontally. Given the focus is to the right of the directrix, it opens to the right.

  2. Find the vertex: The vertex of the parabola is midway between the focus and the directrix. The x-coordinate of the vertex is the average of the x-coordinate of the focus and the x-coordinate of the directrix: \[ x_v = \frac{10 + 2}{2} = 6 \] The y-coordinate of the vertex is the same as the y-coordinate of the focus: \[ y_v = -3 \] So, the vertex is at \( (6, -3) \).

  3. Determine \( p \): The distance \( p \) from the vertex to the focus is: \[ p = 10 - 6 = 4 \] This distance \( p \) is positive since the parabola opens to the right.

  4. Use the standard form: The standard form of a horizontally opening parabola is given by \[ (y - k)^2 = 4p(x - h) \] where \( (h, k) \) is the vertex.

  5. Substituting known values:

    • \( h = 6 \)
    • \( k = -3 \)
    • \( p = 4 \)

    Thus, we substitute: \[ (y + 3)^2 = 16(x - 6) \]

Final Equation:

The equation of the parabola is: \[ (y + 3)^2 = 16(x - 6) \]

From the options given, the correct one that matches our derivation is: \[ \boxed{(y + 3)^2 = 16(x - 6)} \]