MHB Find Equation of Parabola Given Focus & Directrix - Jose's Q&A

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To find the equation of a parabola given the focus at (3,5) and the directrix y=1, the relationship between a point on the parabola, the focus, and the directrix is established. The equation is derived by setting the perpendicular distance from a point (x,y) to the directrix equal to the distance from that point to the focus. This leads to the equation |y-1| = √((x-3)² + (y-5)², which is then squared and simplified. The final equation of the parabola is y = (x² - 6x + 33) / 8. This method effectively demonstrates how to derive a parabola's equation using its focus and directrix.
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Here is the question:

Writing an equation given the directrix and focus?


focus: (3,5) directrix y=1. write an equation for the parabola. How do I do this? please help!

I have posted a link there to this thread so the OP can view my work.
 
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Re: jose's question at Yahoo! Questions: find the equation of the parabola given the focus and direc

Hello jose,

Let's let $(x,y)$ be an arbitrary point on the parabola. Now, we know the perpendicular distance from the point to the directrix will be equal to the distance between this point and the focus. Thus, we may state:

$$|y-1|=\sqrt{(x-3)^2+(y-5)^2}$$

Square both sides:

$$(y-1)^2=(x-3)^2+(y-5)^2$$

Expand binomials:

$$y^2-2y+1=x^2-6x+9+y^2-10y+25$$

Collect like terms:

$$8y=x^2-6x+33$$

Divide through by $8$:

$$y=\frac{x^2-6x+33}{8}$$
 
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