Question

write an equation of the parabola with vertex $(-1,5)$ and focus $(-9,5)$. use the variables $x$ and $y$.
an equation of the parabola is $\square$.

Explanation:

Step1: Determine the form of the parabola

Since the \(y -\)coordinates of the vertex \((-1,5)\) and focus \((-9,5)\) are the same, the parabola opens horizontally. The standard form of a horizontally - opening parabola is \((y - k)^2=4p(x - h)\), where \((h,k)\) is the vertex. Here, \(h=-1,k = 5\).

Step2: Calculate the value of \(p\)

The distance between the vertex \((h,k)=(-1,5)\) and the focus \((h + p,k)=(-9,5)\) is given by \(h + p=-9\). Substitute \(h=-1\) into \(h + p=-9\), we get \(-1+p=-9\), so \(p=-9+1=-8\).

Step3: Substitute \(h,k,p\) into the standard form

Substitute \(h=-1,k = 5,p=-8\) into \((y - k)^2=4p(x - h)\).
We have \((y - 5)^2=4\times(-8)(x+1)\).

Answer:

\((y - 5)^2=-32(x + 1)\)