Question

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

Explanation:

Step1: Recall the standard form of a parabola

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

Step2: Find the value of \(p\)

Given the directrix \(x = 5\) and vertex \((h,k)=(2,-5)\). Using the formula for the directrix \(x=h + p\), substitute \(h = 2\) and \(x = 5\) into \(x=h + p\). So, \(5=2 + p\), which gives \(p=3\).

Step3: Substitute \(h\), \(k\), and \(p\) into the standard - form equation

Substitute \(h = 2\), \(k=-5\), and \(p = 3\) into \((y - k)^2=4p(x - h)\). We get \((y+5)^2=4\times3\times(x - 2)\).

Answer:

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