Question

write the equation of a parabola whose directrix is ( x=-11.5 ) and has a focus at ( (-6.5,-7) ).

Explanation:

Step1: Find the vertex of the parabola

The vertex \((h,k)\) of a parabola is the mid - point between the focus \((x_f,y_f)\) and the directrix \(x = x_d\).
For a parabola with focus \((x_f,y_f)=(-6.5,-7)\) and directrix \(x=-11.5\), the \(y\) - coordinate of the vertex \(k = y_f=-7\).
The \(x\) - coordinate of the vertex \(h=\frac{x_f + x_d}{2}\). Substitute \(x_f=-6.5\) and \(x_d=-11.5\) into the formula: \(h=\frac{-6.5+( - 11.5)}{2}=\frac{-18}{2}=-9\). So the vertex is \((h,k)=(-9,-7)\).

Step2: Find the value of \(p\)

The distance between the vertex \((h,k)\) and the focus \((x_f,y_f)\) (or the vertex and the directrix) is \(p\).
Since the parabola opens to the right (because the focus \(x\) - coordinate \(-6.5\gt\) directrix \(x\) - coordinate \(-11.5\)), \(p=x_f - h\). Substitute \(x_f=-6.5\) and \(h = - 9\) into the formula: \(p=-6.5-( - 9)=2.5\).

Step3: Write the equation of the parabola

The standard form of a parabola that opens to the right is \((y - k)^2=4p(x - h)\).
Substitute \(h=-9\), \(k = - 7\), and \(p = 2.5\) into the formula:
\((y+7)^2=4\times2.5(x + 9)\)
\((y + 7)^2=10(x + 9)\)

Answer:

\((y + 7)^2=10(x + 9)\)