- Define the parabola using the focus-directrix property: \(\sqrt{(x + 5)^2 + (y - 5)^2} = |y + 1|\).
- Square and expand both sides to obtain \((x + 5)^2 + y^2 - 10y + 25 = y^2 + 2y + 1\).
- Simplify the equation by isolating the \(y\) term, resulting in \(12y = (x + 5)^2 + 24\).
- Solve for \(y\) to find the function \(f(x) = \frac{1}{12}(x + 5)^2 + 2\), which corresponds to the last option provided.
- Final answer: \(\boxed{f(x) = \frac{1}{12}(x + 5)^2 + 2}\).
Explanation
- Define the parabola using the focus-directrix definition.
The parabola is defined as the set of all points \((x, y)\) such that the distance to the focus \(F=(-5, 5)\) is equal to the distance to the directrix \(y = -1\).
The distance from a point \((x, y)\) to the focus \((-5, 5)\) is \(\sqrt{(x - (-5))^2 + (y - 5)^2}\).
The distance from a point \((x, y)\) to the line \(y = -1\) is \(|y - (-1)| = |y + 1|\).
Equating these two distances, we have:
\(\sqrt{(x + 5)^2 + (y - 5)^2} = |y + 1|\).
- Algebraically simplify the equation of the parabola.
To eliminate the square root, square both sides of the equation:
\((x + 5)^2 + (y - 5)^2 = (y + 1)^2\).
Expanding the terms involving \(y\):
\((x + 5)^2 + y^2 - 10y + 25 = y^2 + 2y + 1\).
Subtract \(y^2\) from both sides:
\((x + 5)^2 - 10y + 25 = 2y + 1\).
- Solve for \(y\) to express the parabola as a function \(f(x)\).
Group the terms involving \(y\) on one side and the constant terms on the other:
\((x + 5)^2 + 25 - 1 = 10y + 2y\).
\((x + 5)^2 + 24 = 12y\).
Finally, divide by 12 to isolate \(y\):
\(y = \frac{1}{12}(x + 5)^2 + \frac{24}{12}\).
\(y = \frac{1}{12}(x + 5)^2 + 2\).
Thus, the equation of the parabola is \(f(x) = \frac{1}{12}(x + 5)^2 + 2\). Comparing this with the provided options, this matches the fourth choice provided in the prompt.
Examples
Parabolas have a unique property: any light rays or signals traveling parallel to the axis of symmetry and reflecting off the parabola will pass through the focus. This principle is used in designing satellite dishes and car headlights. In a satellite dish, the focus is where the receiver is placed, ensuring that the signals gathered from a large area are concentrated at a single point for processing. Understanding the equation of a parabola helps in determining the optimal shape and focal point for such devices to maximize efficiency.