Question

Write the equation for a parabola with a focus at (1,2)(1,2)left parenthesis, 1, comma, 2, right parenthesis and a directrix at y=6y=6y, equals, 6.
y=

Answer 1

Answer:

  y = -1/8(x -1)² +4

Step-by-step explanation:

The equation of a parabola can be written from focus and directrix using the form ...

  y = 1/(4p)(x -h)² +k

where p is the distance from vertex to focus, and (h, k) is the vertex.

The vertex-focus distance is half the distance from the directrix to the focus. Here, that is ...

  p = (2 -6)/2 = -4/2 = -2

The vertex is halfway between the focus and directrix on the same vertical line, so has coordinates ...

  (h, k) = ((1, 2) +(1, 6))/2 = (1, 4)

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Using these values of p and (h, k), we find the equation of the parabola to be ...

  y = 1/(4(-2))(x -1)² +4

  y = -1/8(x -1)² +4

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Additional comment

You can check the graph by considering the definition of a parabola: each point on the curve is equidistant from the focus and the directrix. This is easily verified for points on the horizontal and vertical lines through the focus. For example, point (5, 2) is 4 units from the focus, and 4 units from the directrix.