Write the equation for a parabola with a focus at (1,-4) and a directrix at x = 2. x=

Question

don't know how to get the equation since it gives me x= 

Answer 1

Since the directrix is vertical instead of horizontal,

the parabola opens sideways instead of up or down.

This means the square term is y instead of x.

 

The basic form of the equation is:

4p(x-h) = (y-k)²

 

(h,k) is the vertex

 

p is the distance from the vertex to either the

directrix or the focus.  The vertex is halfway

between the two.

 

Focus: (1, -4), directrix: x=2

Perpendicular midpoint: (1.5, -4) is vertex

 

Since parabola opens away from the directrix

and toward the focus, it opens left.  This means

p will be negative.... If the parabola opened to

the right then p would be positive.  p = -0.5

 

Putting this together:

4(-0.5)(x-1.5) = (y-(-4))²

-2(x-1.5) = (y+4)²

-2(x-1.5) = y² + 8y + 16

x - 1.5 = (-1/2)(y² + 8y + 16)

x = (-1/2)y² - 4y - 8 + 1.5

x = (-1/2)y² - 4y - 6.5

 

 

 

Answer 2

Jose, that might be a typo.

 

We know that p is the distance from the vertex to both the focus and the directrix, so the vertex is halfway between the focus and the directrix.

 

Since the directrix is vertical, we have a rotated parabola. We also know that the focus is inside the parabola and the directrix is outside, so our parabola opens to the left.

 

Our vertex is at (1.5,-4), and p = .5, so our equation is...

 

(y - (-4))² = -4(.5)(x - 1.5)

(y + 4)² = -2(x - 1.5)

 

Hope this helps!