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Sagot :
Consider we need to find the equation of the parabola.
Given:
Parabola with focus (2, 1) and directrix x=-8.
To find:
The equation of the parabola.
Solution:
We have directrix x=-8. so, it is a horizontal parabola.
The equation of a horizontal parabola is
[tex](y-k)^2=4p(x-h)[/tex] ...(i)
where, (h,k) is center, (h+p,k) is focus and x=h-p is directrix.
On comparing focus, we get
[tex](h+p,k)=(2,1)[/tex]
[tex]h+p=2[/tex] ...(ii)
[tex]k=1[/tex]
On comparing directrix, we get
[tex]h-p=-8[/tex] ...(iii)
Adding (ii) and (iii), we get
[tex]2h=2+(-8)[/tex]
[tex]2h=-6[/tex]
Divide both sides by 2.
[tex]h=-3[/tex]
Putting h=-3 in (ii), we get
[tex]-3+p=2[/tex]
[tex]p=2+3[/tex]
[tex]p=5[/tex]
Putting h=-3, k=1 and p=5 in (i), we get
[tex](y-1)^2=4(5)(x-(-3))[/tex]
[tex](y-1)^2=20(x+3)[/tex]
Therefore, the equation of the parabola is [tex](y-1)^2=20(x+3)[/tex].
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