Question

determine the equation in standard form of the parabola that satisfies the given conditions
directrix at ( x = 5 ); vertex at ( ( 2, - 5 ) )
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Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola with a horizontal axis is \((y - k)^2 = 4p(x - h)\), where \((h,k)\) is the vertex and \(x=h - p\) is the directrix.

Step2: Identify the values from the given information

Given vertex \((h,k)=(2,-5)\) and directrix \(x = 5\). Since the directrix is \(x=h - p\), we have \(5=2 - p\). Solving for \(p\), we get \(p=2 - 5=-3\).

Step3: Substitute into the standard form

Substitute \(h = 2\), \(k=-5\), and \(4p=-12\) into \((y - k)^2 = 4p(x - h)\). The equation becomes \((y+5)^2=-12(x - 2)\)

Answer:

\((y + 5)^2=-12(x - 2)\)