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Anastasy [175]
3 years ago
14

Which equation represents the equation of the parabola with focus (-3 3) and directrix y=7?

Mathematics
1 answer:
Artemon [7]3 years ago
6 0

Answer:

The equation y=\frac{-x^2-6x+31}{8} represents the equation of the parabola with focus (-3, 3) and directrix y = 7.

Step-by-step explanation:

To find the equation of the parabola with focus (-3, 3) and directrix y = 7. We start by assuming a general point on the parabola (x, y).

Using the distance formula d = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 }, we find that the distance between (x, y) is

\sqrt{(x+3)^2+(y-3)^2}

and the distance between (x, y) and the directrix y = 7 is

\sqrt{(y-7)^2}.

On the parabola, these distances are equal so, we solve for y:

\sqrt{(x+3)^2+(y-3)^2}=\sqrt{(y-7)^2}\\\\\left(\sqrt{\left(x+3\right)^2+\left(y-3\right)^2}\right)^2=\left(\sqrt{\left(y-7\right)^2}\right)^2\\\\x^2+6x+y^2+18-6y=\left(y-7\right)^2\\\\x^2+6x+y^2+18-6y=y^2-14y+49\\\\y=\frac{-x^2-6x+31}{8}

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