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jose's question at Yahoo! Answers: find the equation of the parabola given the focus and directrix
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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:
\(\displaystyle |y-1|=\sqrt{(x-3)^2+(y-5)^2}\)
Square both sides:
\(\displaystyle (y-1)^2=(x-3)^2+(y-5)^2\)
Expand binomials:
\(\displaystyle y^2-2y+1=x^2-6x+9+y^2-10y+25\)
Collect like terms:
\(\displaystyle 8y=x^2-6x+33\)
Divide through by $8$:
\(\displaystyle y=\frac{x^2-6x+33}{8}\)
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:
\(\displaystyle |y-1|=\sqrt{(x-3)^2+(y-5)^2}\)
Square both sides:
\(\displaystyle (y-1)^2=(x-3)^2+(y-5)^2\)
Expand binomials:
\(\displaystyle y^2-2y+1=x^2-6x+9+y^2-10y+25\)
Collect like terms:
\(\displaystyle 8y=x^2-6x+33\)
Divide through by $8$:
\(\displaystyle y=\frac{x^2-6x+33}{8}\)