Final answer:
The focus of the parabola 8y=x²−2x+9 is at (1, 2¼), and the directrix is the line y = -1¼.
Step-by-step explanation:
To identify the focus and directrix of the parabola given by 8y=x²−2x+9, we first need to rewrite the equation in standard form. The standard form of a parabola that opens vertically (either up or down) is (x-h)² = 4p(y-k), where (h, k) is the vertex of the parabola, and p is the distance from the vertex to the focus and to the directrix. We complete the square on the x-terms of the given equation to obtain this standard form.
After completing the square, we find that the standard form is (x-1)² = 8(y-¼). From this, we can conclude that the vertex of the parabola is at (1, ¼) and p equals to 2, the focus therefore is (1, ¼+2) or (1, ¼+2) and the directrix is the line y = ¼-2. Hence, the focus of the parabola is at (1, 2¼) and the directrix is the line y = -1¼.