Hey bot, what is the equation of the parabola with the vertex (4,4) and focus (7,4)
The equation of the parabola with vertex (4,4) and focus (7,4) is: (x-4)^2 = 4(3)(y-4) or simplified, x^2 - 8x + 28 = 4y - 16.
To find the equation of a parabola given its vertex and focus, we need to determine the distance between the vertex and the focus, which is called the focal length (denoted as "f"). In this case, the vertex is (4, 4) and the focus is (7, 4), both on the same horizontal line.
Step 1: Find the focal length (f)
The focal length is the distance between the vertex and the focus, which is given by the x-coordinate difference between the two points. In this case, f = |7 - 4| = 3.
Step 2: Determine if the parabola opens to the left or right
Since the focus lies to the right of the vertex, the parabola opens to the right.
Step 3: Write the equation of the parabola in the form (x - h)² = 4p(y - k), where (h, k) represents the vertex coordinates.
The vertex form equation for a parabola is given by (x - h)² = 4p(y - k), where (h, k) represents the vertex coordinates and p represents the focal length.
Plugging in the given vertex coordinates, we have: (x - 4)² = 4p(y - 4).
Step 4: Find the value of p
Since the parabola opens to the right, p = f = 3.
Step 5: Write the final equation of the parabola
(x - 4)² = 4(3)(y - 4).
Simplifying further:
(x - 4)² = 12(y - 4).
Therefore, the equation of the parabola with vertex (4, 4) and focus (7, 4) is (x - 4)² = 12(y - 4).