Final answer:
To find the vertex, focus, and directrix of the parabola given by the equation x - 5 = (y²)², rearrange the equation to the form y = ax + bx². Then, identify the vertex as the point (h, k) and determine the focus by finding the value of p. Finally, find the equation of the directrix by setting it as a horizontal line at y = -p.
Step-by-step explanation:
The given equation, x - 5 = (y²)², can be rearranged to the form y = ax + bx², which represents a parabola. To find the vertex, focus, and directrix of the parabola, we need to rewrite the equation in standard form. Start by isolating the squared term:
(y²)² = x - 5
y² = sqrt(x - 5)
y = ± sqrt(sqrt(x - 5))
From this equation, we can see that the parabola opens upward or downward depending on the sign of the square root. The vertex is the point (h, k) where the parabola reaches its minimum or maximum value. In this case, the vertex is (5, 0) since the term inside the square root is 5.
To find the focus, we need to determine the value of p, which is the distance between the vertex and the focus. The formula for finding p in a parabola of the form y = ax² + bx + c is given by p = 1 / (4a). In this case, a = 0 and the focus is located at a distance of p units vertically from the vertex. Therefore, the focus for this parabola is at the point (5, p).
The directrix is a horizontal line that is equidistant from the vertex as the focus. Since the vertex is at (5, 0), the directrix is a horizontal line at y = -p. Thus, the equation of the directrix is y = -p.