Answer:
Step-by-step explanation:
If you plot the directrix and the focus, you can see that the focus is to the left of the directrix. Since a parabola ALWAYS wraps itself around the focus, our parabola opens sideways, to the left to be specific. The formula for the parabola that opens to the left is
[tex]-(y-k)^2=4p(x-h)[/tex]
We will solve this for x at the end. The negative is out front because it opens to the left. If it opened to the right, it would be positive.
The vertex of a parabola is exactly halfway between the focus and the directrix, so our vertex coordinates h and k are (3, 6). P is defined as the distance between the vertex and the directrix, or the vertex and the focus. Since the vertex is directly between both the directrix and the focus, each distance is the same. P = 1. Filling in what we have now:
[tex]-(y-6)^2=4(1)(x-3)[/tex] which simplifies to
[tex]-(y-6)^2=4(x-3)[/tex]
Now we will solve it for x.
[tex]-(y-6)^2=4x-12[/tex] and
[tex]-(y-6)^2+12=4x[/tex] so
[tex]-\frac{1}{4}(y-6)^2+12=x[/tex]
The equation of a parabola is required from the given directric and focus.
The equation of the parabola is [tex]x=-\dfrac{y^2}{4}+3y-6[/tex]
Directrix = [tex]x=4[/tex]
Focus = [tex](2,6)[/tex]
The equation of parabola is given by
[tex](x-2)^2+(y-6)^2=(x-4)^2\\\Rightarrow x^2+4-4x+y^2+36-12y=x^2+16-8x\\\Rightarrow 40-4x+y^2-12y=16-8x\\\Rightarrow 40-4x+y^2-12y-16+8x=0\\\Rightarrow 24+4x+y^2-12y=0\\\Rightarrow 4x=-y^2+12y-24\\\Rightarrow x=-\dfrac{y^2}{4}+3y-6[/tex]
The equation of the parabola is [tex]x=-\dfrac{y^2}{4}+3y-6[/tex].
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