The equation for the parabola is [tex]x^{2} -8x-6y+1=0[/tex].
So, it is given in the question that [tex]focus(4,-1)[/tex] and directrix,[tex]y=-4[/tex].
We have to find the equation of the parabola,
So, as we know the equation of the parabola that [tex]x^{2} =4ay[/tex]
But if a parabola has a vertical axis the standard form of the equation of the parabola is:
[tex](x-h)^{2}=4a(y-k)[/tex]
and the vertex of this parabola is at [tex](h,k)[/tex].
The focus is at [tex](h,k+a)[/tex] and the directrix is the line [tex]y=k-a[/tex].
Here, the parabola opens upward, so [tex]a > 0[/tex]
So, considering all the above criteria, we get;
vertex of this parabola is at [tex](4,-\frac{5}{2} )[/tex] and [tex]a=\frac{3}{2}[/tex]
Now, putting the above-calculated value in the standard equation of the parabola, we get;
[tex](x-4)^{2}= 4*\frac{3}{2}*(y-(-\frac{5}{2} ))[/tex]
after solving the above equation, we get;
[tex]x^{2} -8x-6y+1=0[/tex]
Hence, [tex]x^{2} -8x-6y+1=0[/tex] is the equation for a parabola.
What is the parabola?
A plane curve generated by a point moving so that its distance from a fixed point is equal to its distance from a fixed line : the intersection of a right circular cone with a plane parallel to an element of the cone. something bowl-shaped (such as an antenna or microphone reflector)
What is formula of parabola?
The equation of a parabola can be written in two basic forms: Form 1: y = a( x – h) 2 + k. Form 2: x = a( y – k) 2 + h.
To learn more about parabola, refer to:
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