Question

1. focus (4,-2), directrix y = -4
2. focus (1,-9), directrix x = -5

focal width vertex
focal width vertex
equation
equation

Explanation:

Step1: Calculate vertex for problem 7

The vertex \((h,k)\) of a parabola is the mid - point between the focus \((h,k + p)\) and the directrix \(y=k - p\). For focus \((4,-2)\) and directrix \(y = - 4\), \(h = 4\), \(k=\frac{-2+( - 4)}{2}=\frac{-2 - 4}{2}=-3\). So the vertex is \((4,-3)\).

The distance between the focus and the vertex \(p=-2-(-3) = 1\). The focal width \(=4|p|=4\times1 = 4\).

The standard form of a parabola that opens up or down is \((x - h)^{2}=4p(y - k)\). Substituting \(h = 4\), \(k=-3\), \(p = 1\) into the formula, we get \((x - 4)^{2}=4(y + 3)\).

Step2: Calculate vertex for problem 8

The vertex \((h,k)\) of a parabola is the mid - point between the focus \((h + p,k)\) and the directrix \(x=h - p\). For focus \((1,-4)\) and directrix \(x=-5\), \(k=-4\), \(h=\frac{1+( - 5)}{2}=\frac{1-5}{2}=-2\). So the vertex is \((-2,-4)\).

The distance between the focus and the vertex \(p=1-(-2)=3\). The focal width \(=4|p|=4\times3 = 12\).

The standard form of a parabola that opens left or right is \((y - k)^{2}=4p(x - h)\). Substituting \(h=-2\), \(k = - 4\), \(p = 3\) into the formula, we get \((y + 4)^{2}=12(x + 2)\).

Answer:

For problem 7:

• Focal width: \(4\)
• Vertex: \((4,-3)\)
• Equation: \((x - 4)^{2}=4(y + 3)\)

For problem 8:

• Focal width: \(12\)
• Vertex: \((-2,-4)\)
• Equation: \((y + 4)^{2}=12(x + 2)\)