Answer:
The equation of the parabola with a focus at (-5,5) and a directrix of y = -1 is [tex]y = \frac{1}{12}\cdot (x+5)^{2}+2[/tex].
Step-by-step explanation:
From statement we understand that parabola has its axis of symmetry in an axis parallel to y-axis. According to Analytical Geometry, the minimum distance between focus and directrix equals to twice the distance between vertex and any of endpoints.
If endpoints are (-5, 5) and (-5, -1), respectively, then such distance ([tex]r[/tex]), dimensionless, is calculated by means of the Pythagorean Theorem:
[tex]r = \frac{1}{2}\cdot \sqrt{[-5-(-5)]^{2}+[5-(-1)]^{2}}[/tex]
[tex]r = 3[/tex]
And the location of the vertex ([tex]V(x,y)[/tex]), dimensionless, which is below the focus, is:
[tex]V(x,y) = F(x,y)-R(x,y)[/tex] (1)
Where:
[tex]F(x,y)[/tex] - Focus, dimensionless.
[tex]R(x,y)[/tex] - Vector distance, dimensionless.
If we know that [tex]F(x,y) = (-5,5)[/tex] and [tex]R(x,y) = (0,3)[/tex], then the location of the vertex is:
[tex]V(x,y) = (-5,5)-(0,3)[/tex]
[tex]V(x,y) =(-5,2)[/tex]
In addition, we define a parabola by the following expression:
[tex]y-k = \frac{(x-h)^{2}}{4\cdot r}[/tex] (2)
Where:
[tex]h[/tex], [tex]k[/tex] - Coordinates of the vertex, dimensionless.
[tex]r[/tex] - Distance of the focus with respect to vertex, dimensionless.
If we know that [tex]h = -5[/tex], [tex]k = 2[/tex] and [tex]r = 3[/tex], then the equation of the parabola is:
[tex]y = \frac{1}{12}\cdot (x+5)^{2}+2[/tex]
The equation of the parabola with a focus at (-5,5) and a directrix of y = -1 is [tex]y = \frac{1}{12}\cdot (x+5)^{2}+2[/tex].
