Answer:
The equation of the parabola with a focus at (-5,5) and a directrix of y = -1 is .
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 (), dimensionless, is calculated by means of the Pythagorean Theorem:
And the location of the vertex (), dimensionless, which is below the focus, is:
(1)
Where:
- Focus, dimensionless.
- Vector distance, dimensionless.
If we know that and
, then the location of the vertex is:
In addition, we define a parabola by the following expression:
(2)
Where:
,
- Coordinates of the vertex, dimensionless.
- Distance of the focus with respect to vertex, dimensionless.
If we know that ,
and
, then the equation of the parabola is:
The equation of the parabola with a focus at (-5,5) and a directrix of y = -1 is .