Answer:
So, the correct answer is:
Focus at (3, 0.875); directrix at y = 1.125
Explanation:
The equation of the parabola is given as 2(y + 1)² + 3. To determine the focus and directrix of the parabola, we need to express the equation in the standard form (y - k)² = 4a(x - h), where (h, k) represents the vertex of the parabola.
1. Convert the equation to the standard form:
2(y + 1)² + 3 = 4a(x - h)
Comparing this equation with the standard form, we can see that h = 0 and k = -1.
2. Determine the value of 'a':
To find the value of 'a', we need to rewrite the equation in the standard form by completing the square. Let's do that:
2(y + 1)² + 3 = 4a(x - h)
2(y + 1)² = 4a(x - h) - 3
Dividing both sides by 2, we get:
(y + 1)² = 2a(x - 0) - 3/2
(y + 1)² = 2a(x - 0) - 1.5
Comparing this equation with the standard form (y - k)² = 4a(x - h), we can see that h = 0 and k = -1, and the coefficient of (x - h) is 2a.
Since 2a = 2, we can conclude that 'a' is equal to 1.
3. Find the focus and directrix:
The focus of the parabola is given by the coordinates (h + a, k), and the directrix is a vertical line with the equation x = h - a.
Plugging in the values, we have:
Focus: (h + a, k) = (0 + 1, -1) = (1, -1)
Directrix: x = h - a = 0 - 1 = -1
So, the correct answer is:
Focus at (3, 0.875); directrix at y = 1.125