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nikklg [1K]
3 years ago
8

Find the equation of the parabola with focus (5, 1) and directrix y = -1.

Mathematics
1 answer:
Mumz [18]3 years ago
8 0
Check the picture below.

since the focus point is above the directrix, that simply means the parabola is a vertical one, and therefore the square variable is the "x".

keeping in mind that, there's a distance "p" from the vertex to either the focus point or the directrix, that puts the vertex half-way between those fellows, in this case at 0, right between 1 and -1, as you see in the picture, and the parabola looks like so.

since the parabola is opening upwards, the value for "p" is positive, thus

\bf \textit{parabola vertex form with focus point distance}
\\\\
\begin{array}{llll}
4p(x- h)=(y- k)^2
\\\\
\boxed{4p(y- k)=(x- h)^2}
\end{array}
\qquad 
\begin{array}{llll}
vertex\ ( h, k)\\\\
 p=\textit{distance from vertex to }\\
\qquad \textit{ focus or directrix}
\end{array}\\\\
-------------------------------\\\\
\begin{cases}
h=5\\
k=0\\
p=1
\end{cases}\implies 4(1)(y-0)=(x-5)^2
\\\\\\
4y=(x-5)^2\implies  y=\cfrac{1}{4}(x-5)^2

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<span><span>(-3,1). y=mx+b or 1=-3/5 × -3+b, or solving for b: b=1-(-3/5)(-3). b=-4/5.</span><span>(7,-5). y=mx+b or -5=-3/5 × 7+b, or solving for b: b=-5-(-3/5)(7). b=-4/5.</span></span>See! In both cases we got the same value for b. And this completes our problem.

<span><span>The equation of the line that passes through the points(-3,1) and (7,-5)is</span><span>y=-3/5x-4/5"</span>
Source: webmath</span>
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