again, find the equation of the parabola with the given fetures.
focus (0,2) directrix x=2
please explain step by step if possible!
thank you
Definitnion:A parabola is a set of point that are equidistant from a given point (focus) and a given line (directrix)Originally Posted by mattballer082
Let $\displaystyle (x,y)$ be a point on the parabola. Then it is equidistant from the focus $\displaystyle (0,2)$ and the line, $\displaystyle x=2$. The distance to the focus by the distance formula is,
$\displaystyle s=\sqrt{x^2+(y-2)^2}$
Now, what is distance to the line $\displaystyle x=2$. This is slightly difficult. To find the distance from a point to a line you need to draw a perpendicular line, then find the distance of that line. Note, the $\displaystyle x=2$ is a vertical line, that means if you draw (and try to imagine this) a perpendicular line from the point $\displaystyle (x,y)$ you intersect at $\displaystyle (2,y)$. Now the distance between these two points is,
$\displaystyle s=\sqrt{(2-x)^2+(y-y)^2}=\sqrt{(2-x)^2}$
By, definition these two distances are equal hence,
$\displaystyle \sqrt{x^2+(y-2)^2}=\sqrt{(2-x)^2}$
Square both sides,
$\displaystyle x^2+(y-2)^2=(2-x)^2$
Thus,
$\displaystyle x^2+y^2-4y+4=4-4x+x^2$
Kill the unneccesarry,
$\displaystyle y^2-4y=-4x$
Rewrite as,
$\displaystyle y^2+4x-4y=0$
Hello, mattballer0821
Find the equation of the parabola with the given features.
Focus (0,2) directrix x=2
You're expected to know these basic facts . . .
The Vertex is halfway between the Focus and the Directrix.
There are two forms:
. . . . Vertical: $\displaystyle (x-h)^2 = 4p(y-k)$ . . . opens up or down: $\displaystyle \cup$ or $\displaystyle \cap$
. . Horizontal: $\displaystyle (y-k)^2 = 4p(x-h)$ . . . opens right or left: $\displaystyle \subset$ or $\displaystyle \supset$
where $\displaystyle (h,k)$ is the Vertex of the parabola
and $\displaystyle p$ is the directed distance from the Vertex to the Focus.
The parabola always "bends around" the Focus.
The parabola always "bends away" from the Directrix.
Sketch the given information:Code:o | |o : | o : F(0,2)* oV : | o : - - - - +o- - - + - o | : | -x=2
We see that the Vertex is $\displaystyle (1,2)\quad\Rightarrow\quad (h,k) = (1,2)$
. . the parabola opens to the left: $\displaystyle (y - k)^2 = 4p(x - h)$
. . distance from Vertex to Focus is $\displaystyle 1$ to the left: $\displaystyle p = -1$
Therefore: .$\displaystyle (y - 2)^2 \,= \,-4(x - 1)$