Question

1. (02.08) find the equation of the parabola with a focus at (-5, -5) and a directrix of y = 7. (2 points) f(x)=-\frac{1}{24}(x - 1)^2-5 f(x)=\frac{1}{24}(x - 1)^2-5 f(x)=-\frac{1}{24}(x + 5)^2+1 f(x)=\frac{1}{24}(x + 5)^2+1

Explanation:

Step1: Recall parabola formula

The standard - form of a parabola with a vertical axis of symmetry is \((x - h)^2=4p(y - k)\), where \((h,k)\) is the vertex and \(p\) is the distance between the vertex and the focus (or the vertex and the directrix). The vertex is the mid - point between the focus \((x_f,y_f)=(-5,-5)\) and the point on the directrix directly above (or below) the focus. The directrix is \(y = 7\).

Step2: Calculate the vertex

The \(x\) - coordinate of the vertex \(h=-5\). The \(y\) - coordinate of the vertex \(k=\frac{-5 + 7}{2}=1\). So the vertex is \((h,k)=(-5,1)\).

Step3: Calculate the value of \(p\)

The distance \(p\) between the vertex \((-5,1)\) and the focus \((-5,-5)\) is \(p=-5 - 1=-6\) (or the distance between the vertex and the directrix \(7 - 1 = 6\), and since the focus is below the directrix, \(p=-6\)).

Step4: Write the equation of the parabola

Starting with \((x - h)^2=4p(y - k)\), substituting \(h=-5\), \(k = 1\), and \(p=-6\), we get \((x+5)^2=4\times(-6)(y - 1)\), which simplifies to \(y=-\frac{1}{24}(x + 5)^2+1\).

Answer:

\(f(x)=-\frac{1}{24}(x + 5)^2+1\)