Question

question 9
write the equation of a parabola whose directrix is ( x = -2 ) and has a focus at ( (-4, -7) ).
question help:
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written example

Explanation:

Step1: Find the vertex

The vertex \((h,k)\) of a parabola is halfway between the directrix \(x = -2\) and the focus \((-4,-7)\). For the \(x\)-coordinate of the vertex, we use the midpoint formula: \(h=\frac{-2 + (-4)}{2}=\frac{-6}{2}=-3\). The \(y\)-coordinate of the vertex is the same as the \(y\)-coordinate of the focus, so \(k = -7\). So the vertex is \((-3,-7)\).

Step2: Determine the direction and \(p\)

Since the directrix is a vertical line (\(x=-2\)) and the focus is to the left of the directrix, the parabola opens to the left. The distance between the vertex and the directrix (or focus) is \(|p|\). The distance between \(x=-3\) (vertex \(x\)-coordinate) and \(x=-2\) (directrix) is \(|-2-(-3)| = 1\). Since it opens to the left, \(p=-1\) (negative because it's in the direction opposite to the opening for horizontal parabolas).

Step3: Write the equation

The standard form of a horizontal parabola (opening left or right) is \((y - k)^2=4p(x - h)\). Substituting \(h=-3\), \(k = -7\), and \(p=-1\) into the formula: \((y - (-7))^2=4(-1)(x - (-3))\), which simplifies to \((y + 7)^2=-4(x + 3)\).

Answer:

\((y + 7)^2=-4(x + 3)\)