To derive the equation of a parabola with a given focus and directrix, we can use the definition of a parabola: it is the set of all points that are equidistant from the focus and the directrix.
In this case:
- The focus is at \( (6, -2) \).
- The directrix is at \( y = 0 \).
Let \( P(x, y) \) be any point on the parabola. The distance from point \( P \) to the focus \( (6, -2) \) is given by:
\[ d(P, \text{focus}) = \sqrt{(x - 6)^2 + (y + 2)^2} \]
The distance from point \( P \) to the directrix \( y = 0 \) is the vertical distance from the point to the line:
\[ d(P, \text{directrix}) = |y - 0| = |y| \]
Setting these two distances equal gives us:
\[ \sqrt{(x - 6)^2 + (y + 2)^2} = |y| \]
Next, we square both sides to eliminate the square root:
\[ (x - 6)^2 + (y + 2)^2 = y^2 \]
Now, expand the left side:
\[ (x - 6)^2 + (y^2 + 4y + 4) = y^2 \]
Simplify by cancelling \( y^2 \) from both sides:
\[ (x - 6)^2 + 4y + 4 = 0 \]
Now, rearranging gives:
\[ 4y = -(x - 6)^2 - 4 \]
Divide by 4 to express \( y \) in standard form:
\[ y = -\frac{1}{4}(x - 6)^2 - 1 \]
To write the equation in a more standard form, we can expand it:
\[ y = -\frac{1}{4}(x^2 - 12x + 36) - 1 \]
Distributing \(-\frac{1}{4}\):
\[ y = -\frac{1}{4}x^2 + 3x - 9 - 1 \] \[ y = -\frac{1}{4}x^2 + 3x - 10 \]
Among the options provided, the correct equation of the parabola is:
\[ y = -\frac{1}{4}x^2 + 3x - 10 \]