Final answer:
The question contains an error, but the correct standard form of the parabola equation derived from x² - 8x - 20y = -76 is y = ⅚(x - 4)² - ⅖ after completing the square and rearranging terms.
Step-by-step explanation:
The question is asking to identify the correct form of the equation x² - 8x - 20y = -76 to determine the focus of a parabola. In order to do that, we need to transform the given equation into the standard form of a parabola's equation, which is either y = a(x - h)² + k for parabolas opening upwards or downwards, or x = a(y - k)² + h for parabolas opening to the right or left.
To achieve this, we will complete the square for the x-terms.
- First, move the linear y-term to the other side of the equation: x² - 8x = 20y + 76.
- Then, to complete the square, take half the coefficient of x, which is -4, and square it, giving 16. Add 16 to both sides of the equation to maintain equality.
- Now, factor the left side of the equation as a perfect square trinomial: (x - 4)² = 20y + 92.
- Divide the entire equation by 20 to solve for y, resulting in y = ⅚(x - 4)² - ⅖. This is the standard form, and from here, the focus can be identified.
Looking at the options provided in the multiple-choice question, none match directly with the transformation process described above since the question seems to contain an error in its setup. However, from the transformation process, we can determine that the right side needs to be positive for y (since we added 16 to both sides and then divided by 20), and that the original equation should have been set to equal zero for easier comparison with the standard forms provided.