The equations that represent a parabola with the focus (0,4) and the vertex (0, -6) are 40y = [tex]x^{2}[/tex] - 240 and [tex]x^{2}[/tex] = 40 (y + 6)
What is parabola?
A parabola refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed line. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola.
The general equation of a parabola is:
[tex](y -k)^{2}[/tex] = 4a(x − h) for horizontal parabola
[tex](x - h) ^{2}[/tex]= 4a(y − k) for vertical parabola
where (h,k) denotes the vertex
a = point of focus
According to question
focus of parabola :(0,4)
vertex of parabola (h,k): (0, -6)
now, form points given
a = 4 − (-6) = 10 as x coordinates are the same.
Since the focus lies to the left of vertex, a = 10
By The general equation of a parabola :
[tex](x - h) ^{2}[/tex]= 4a(y − k) for vertical parabola
[tex](x - 0)^{2}[/tex] = 4*10(y-(-6))
[tex]x^{2}[/tex] = 40 (y + 6) --------------------- (1)
or
[tex]x^{2}[/tex] = 40y + 240
40y = [tex]x^{2}[/tex] - 240 ---------------------------(2)
Therefore , parabola can be represented by equation (1) and (2).
Hence, the equations that represent a parabola with the focus (0,4) and the vertex (0, -6) are 40y = [tex]x^{2}[/tex] - 240 and [tex]x^{2}[/tex] = 40 (y + 6)
To know more about parabola here:
https://brainly.com/question/4074088
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