Final Answer:
The line passing through (0, -1) and having a positive slope could pass through points (12,3), (-3, 1), and (1, 15). Thus the correct options are A, C and D.
Step-by-step explanation:
Given point:
(P(0, -1)
Equation of a line in point-slope form:
(y - y₁) = m(x - x₁), where (x₁, y₁) = (0, -1) and m represents the positive slope
Given positive slope:
Let's assume m = k (a positive value).
Now, let's test each point:
A (12, 3):
Substitute (x₁, y₁) = (0, -1) and the coordinates of point A into the point-slope equation:
3 - (-1) = k(12 - 0)
4 = 12k
k = 4/12 = 1/3
C (-3, 1):
Substitute (x₁, y₁) = (0, -1) and the coordinates of point C into the point-slope equation:
1 - (-1) = k(-3 - 0)
2 = -3k
k = 2/-3 = -2/3
D (1, 15):
Substitute (x₁, y₁) = (0, -1) and the coordinates of point D into the point-slope equation:
15 - (-1) = k(1 - 0)
16 = k
For points B (-2, -5) and E (5, -2), let's follow the same process:
B (-2, -5):
y - (-1) = k(-2 - 0)
-5 + 1 = -2k
-4 = -2k
k = -4/-2= 2
E (5, -2):
y - (-1) = k(5 - 0)
-2 + 1 = 5k
-1 = 5k
k = -1/5
From these calculations, the values of k for points A, C, and D are k = 1/3, k = -2/3, and k = 16 respectively. Only points A, C, and D result in positive or negative slopes as required, indicating that the line passing through the point (0, -1) with a positive slope can go through points A (12, 3), C (-3, 1), and D (1, 15).
Therefore, the correct options are A, C and D.