Answer
3) Option C is correct.
(x + 6)² = 4 (y - 2)
4) Option B is correct.
(y - 3)² = 8 (x + 1)
Step-by-step explanation
A parabola with a vertical axis will have a standard equation of the parabola as
(x - h)² = 4p (y - k),
where p ≠ 0.
The vertex of this parabola is at (h, k). The focus is at (h, k + p).
The directrix is the line y = k - p and it is a horizontal directrix.
A parabola with a horizontal axis will have a standard equation of the parabola as
(y - k)² = 4p (x - h),
where p ≠ 0.
The vertex of this parabola is at (h, k). The focus is at (h + p, k).
The directrix is the line x = h - p and it is a vertical directrix.
So, for question 3
focus = (-6, 3)
Directrix : y = 1
From the directrix information, we can tell that it is a horizontal directrix and the parabola has a vertical axis.
The parabola equation is thus
(x - h)² = 4p (y - k),
focus = (h, k + p) = (-6, 3)
h = -6, k + p = 3
directrix : y = k - p = 1
So,
k + p = 3
k - p = 1
Add these two and we have
2k = 4
(2k/2) = (4/2)
k = 2
k + p = 3
2 + p = 3
p = 3 - 2 = 1
h = -6, p = 1, k = 2
(x - h)² = 4p (y - k),
(x - -6)² = 4(1) (y - 2)
(x + 6)² = 4 (y - 2)
Option C is correct.
For question 4,
focus = (1, 3)
Directrix : x = -3
From the directrix information, we can tell that it is a vertical directrix and the parabola has a horizontal axis.
The parabola equation is thus
(y - k)² = 4p (x - h),
focus = (h + p, k) = (1, 3)
h + p = 1, k = 3
directrix : x = h - p = -3
So,
h + p = 1
h - p = -3
Add these two and we have
2h = -2
(2h/2) = (-2/2)
h = -1
h + p = 1
-1 + p = 1
p = 1 + 1 = 2
h = -1, p = 2, k = 3
(y - k)² = 4p (x - h)
(y - 3)² = 4(2) (x - -1)
(y - 3)² = 8 (x + 1)
Option B is correct.
Hope this Helps!!!