Answer:
The statements that can be made about the parabola are;
The parabola opens towards the right
The equation of the parabola is (y - 1)² = 16 × (x + 1)
The parabola has a vertex at (-1, 1)
Step-by-step explanation:
The given focus of the parabola is (3, 1)
The directrix of the parabola, x = -5
Therefore, from the location of the directrix (on the x-axis) and the location of the focus relative to the directrix towards the positive direction of the x-axis relative to the directrix, the parabola opens towards the left of the coordinate chart
The general equation of the focus = (h + p, k)
The general equation of the directrix, x = h - p
Comparing with the values of the focus and the directrix of the given parabola, we have;
(3, 1) = (h + p, k)
-5 = h - p
Therefore, we get;
k = 1
h + p = 3...(1)
h - p = -5...(2)
Adding equation (1) to equation (2) gives;
h + h + p - p = 3 + (-5)
2·h = -2
h = -2/2 = -1
h = -1
From equation (1), we get;
h + p = -1 + p = 3
∴ p = 3 + 1 = 4
p = 4
The vertex of the parabola, (h, k) = (-1, 1)
The equation of the parabola in the form (y - k)² = 4·p·(x - h) is therefore, presented as follows;
The equation of the parabola = (y - 1)² = 4 × 4 × (x - (-1))
(y - 1)² = 16 × (x - (-1))
The equation of the parabola is (y - 1)² = 16 × (x + 1).
