Question

1. find the equation of the parabola with the focus at (-3,1) and vertex at (-1,1).

Explanation:

Step1: Determine the form of the parabola

Since the y - coordinate of the focus \((-3,1)\) and the vertex \((-1,1)\) is the same, the parabola is of the form \((y - k)^{2}=4p(x - h)\), where \((h,k)\) is the vertex. Here \(h=-1,k = 1\).

Step2: Calculate the value of \(p\)

The distance between the vertex \((h,k)\) and the focus \((h + p,k)\) is given by \(p\). We know that \(h=-1\) and the x - coordinate of the focus is \(x=-3\). Using the formula \(h + p=-3\), substituting \(h=-1\) gives \(-1+p=-3\), so \(p=-2\).

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

Substitute \(h=-1,k = 1,p=-2\) into \((y - k)^{2}=4p(x - h)\). We get \((y - 1)^{2}=4\times(-2)(x+1)\)

Answer:

\((y - 1)^{2}=-8(x + 1)\)