Answer:
The equation [tex]y=\frac{-x^2-6x+31}{8}[/tex] 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 [tex]d = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 }[/tex], we find that the distance between (x, y) is
[tex]\sqrt{(x+3)^2+(y-3)^2}[/tex]
and the distance between (x, y) and the directrix y = 7 is
[tex]\sqrt{(y-7)^2}[/tex].
On the parabola, these distances are equal so, we solve for y:
[tex]\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}[/tex]
The equation of the parabola is (x + 3)² = -8(y - 5)
Parabola is the locus of a point such that the same distance from a fixed line , called the directrix , and a fixed point (the focus) is the same.
The equation of a parabola is:
(x - h)² = 4p(y - k);
The directrix is at: y = k - p, focus is at (h, k + p)
Given focus (-3, 3) and directrix y=7, hence:
h = -3
k + p = 3 (1)
k - p = 7 (2)
From equation 1 and 2, k = 5 and p = -2
The equation of the parabola is (x + 3)² = -8(y - 5)
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