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Sagot :
The equation of the parabola may be written as (x-3)²=-8(y-6) or y = -1/8(x²-6x-39)
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Since the x coordinates of the vertex (3, 6) and the focus (3, 4) are equal, this implies that the parabola is vertical. If the parabola is vertical, then we will use the equation:
[tex](x-h)^2=4p(y-k)[/tex]
where:
- [tex]h[/tex] and [tex]k[/tex] are the x-coordinate and y-coordinate of the vertex
- [tex](h,k+p)[/tex] refers to the coordinates of the focus.
In our problem, we have the following:
[tex](h,k)=(3,6) \quad \quad (h,k+p)=(3,4)[/tex]
Solve for [tex]p[/tex] using the information above, we can get that
[tex]k+p=4[/tex]
[tex]6+p=4[/tex]
[tex]\therefore p=4-6=-2[/tex]
Now that we found [tex]p[/tex] we can find the general equation. Recall that we use the general equation of the parabola:
[tex](x-h)^2=4p(y-k)[/tex]
Substitute the values and simplify, to get:
[tex](x-3)^2=4(-2)(y-6)[/tex]
[tex]\boxed{(x-3)^2=-8(y-6)}[/tex]
The equation above may be the answer, but if you want to solve for [tex]y [/tex] then:
[tex]x^2-6x+9=-8y+48[/tex]
[tex]x^2-6x-39=-8y[/tex]
[tex]\boxed{\displaystyle y=-\frac{1}{8}(x^2-6x-39)}[/tex]
Therefore, the equation of the parabola may be written as y = -1/8(x²-6x-39).
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Hope it helps.
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