Question

assignment 10.3: the parabola
score: 58.35/100 answered: 3/5
question 4
write the equation of a parabola whose directrix is y = 0 and has a focus at (3, - 10).
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Explanation:

Step1: Find the vertex of the parabola

The vertex \((h,k)\) is the mid - point between the focus \((x_f,y_f)=(3,-10)\) and the directrix \(y = 0\).
The formula for the \(y\) - coordinate of the vertex \(k=\frac{y_f + y_d}{2}\), where \(y_d\) is the \(y\) - coordinate of the directrix. Here \(y_f=-10\) and \(y_d = 0\), so \(k=\frac{-10 + 0}{2}=-5\). The \(x\) - coordinate of the vertex \(h\) is the same as the \(x\) - coordinate of the focus, so \(h = 3\). The vertex is \((h,k)=(3,-5)\).

Step2: Find the value of \(p\)

The distance between the vertex \((h,k)\) and the focus \((h,k + p)\) (or the vertex and the directrix). Since the vertex is \((3,-5)\) and the focus is \((3,-10)\), then \(k + p=-10\). Substituting \(k=-5\) into \(k + p=-10\), we get \(-5+p=-10\), so \(p=-5\).

Step3: Use the standard form of the parabola equation

The standard form of a parabola that opens up or down is \((x - h)^2=4p(y - k)\). Substituting \(h = 3\), \(k=-5\) and \(p=-5\) into the equation:

$$
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$$

Answer:

\((x - 3)^2=-20(y + 5)\)