What is an equation of a parabola with the given vertex and focus?
Vertex: (5, 4)
Focus: (8, 4)
a. x = 1/2 (y - 4)^2 + 5
b. y = 1/2 (x + 4)^2 - 5
c. x = 1/2 (y + 4)^2 - 5
d. y = 1/2 (x - 4)^2 + 5
form (opens right)
(y-4)^2 = 4 a (x-5)
a is distance from vertex to focus = 8-5 = 3
so
(y-4)^2 = 12 (x-5) = 12 x - 60
12 x = (y-4)^2 + 60
x = (1/12) (y-4)^2 + 5
like a. but 1/12 not 1/2 Typo?
To determine the equation of a parabola given its vertex and focus, we can use the standard form of a parabola equation:
(x-h)^2 = 4p(y-k)
Where (h, k) represents the coordinates of the vertex, and p is the distance between the focus and the vertex.
In this case, the given vertex is (5, 4) and the focus is (8, 4).
Step 1: Find the value of p by calculating the distance between the vertex and the focus.
Distance Formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((8 - 5)^2 + (4 - 4)^2)
d = sqrt(3^2 + 0)
d = sqrt(9)
d = 3
Therefore, p = 3.
Step 2: Substitute the values of (h, k) and p into the standard form equation.
(x-5)^2 = 4(3)(y-4)
Simplifying further, we obtain:
(x-5)^2 = 12(y-4)
Hence, the correct equation of the parabola is:
(x-5)^2 = 12(y-4)
Comparing this equation to the given options:
a. x = 1/2 (y - 4)^2 + 5
b. y = 1/2 (x + 4)^2 - 5
c. x = 1/2 (y + 4)^2 - 5
d. y = 1/2 (x - 4)^2 + 5
We can see that option d is the correct equation that matches our findings. Therefore, the answer is:
d. y = 1/2 (x - 4)^2 + 5