Question

skylar is designing a flashlight that uses a parabolic reflecting mirror and a light source. the shape of the mirror can be modeled by $(y + 5)^2=12(x + 2)$, where $x$ and $y$ are measured in inches. what is the focus of the flashlight? (-5, 1) (-2, -5) (1, -5) (5, 1)

Explanation:

Step1: Identify the form of the parabola equation

The given equation is \((y + 5)^2=12(x + 2)\), which is in the form \((y - k)^2 = 4p(x - h)\) for a parabola that opens to the right or left. Here \(h=-2\), \(k = - 5\) and \(4p=12\).

Step2: Solve for \(p\)

Since \(4p = 12\), then \(p=\frac{12}{4}=3\).

Step3: Find the focus

For a parabola of the form \((y - k)^2=4p(x - h)\), the focus is given by the point \((h + p,k)\). Substituting \(h=-2\), \(p = 3\) and \(k=-5\) into the formula, we get \((-2+3,-5)=(1,-5)\).

Answer:

C. \((1,-5)\)