1. Understand the problem:
We need to find the equation of a parabola given its focus at (6, 0) and directrix \( x = -6 \).
2. Recall the definition of a parabola:
A parabola is the locus of points equidistant from the focus and the directrix.
3. Let \( (x, y) \) be any point on the parabola:
The distance from \( (x, y) \) to the focus (6, 0) is:
\[ \sqrt{(x - 6)^2 + (y - 0)^2} \]
The distance from \( (x, y) \) to the directrix \( x = -6 \) is:
\[ |x + 6| \]
4. Set the distances equal and simplify:
\[ \sqrt{(x - 6)^2 + y^2} = |x + 6| \]
Square both sides to eliminate the square root and absolute value:
\[ (x - 6)^2 + y^2 = (x + 6)^2 \]
Expand both sides:
\[ x^2 - 12x + 36 + y^2 = x^2 + 12x + 36 \]
Simplify by canceling \( x^2 \) and 36 from both sides:
\[ -12x + y^2 = 12x \]
Combine like terms:
\[ y^2 = 24x \]
Correct Answer: (A) \( y^2 = 24x \)
The focus is $(6, 0)$, and the directrix is $x = -6$.
The vertex is midway between the focus and directrix: Vertex = $\left(\frac{6 + (-6)}{2}, 0\right) = (0, 0)$.
The parabola opens to the right.
The equation of the parabola is: $y^2 = 4ax$, where $a = 6$.
Substituting $a = 6$, we get: $y^2 = 24x$.