SOLUTION: find equation of parabola whose axis is x+2=0,directrix is y-4=0,latus rectum=6 and concativity towards positive direction of y axis?

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Question 154471: find equation of parabola whose axis is x+2=0,directrix is y-4=0,latus rectum=6 and concativity towards positive direction of y axis?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Find equation of parabola whose axis is x+2=0,
directrix is y-4=0, latus rectum=6 and concativity
towards positive direction of y-axis?

Let's draw the axis of symmetry, x%2B2=0

 

The vertex and the focus must be on this line of
symmetry.  So their x-coordinates must be -2

Now we'll draw the directrix in green:

 

The equation of a parabola which is concave upward
or downward is

%28x-h%29%5E2=4p%28y-k%29

(h,k) is the vertex

p = directed distance from directrix to vertex = 
distance from vertex to focus.

Here we use the fact that the latus rectum
has length abs%284p%29.

We are told the the latus rectum has length 6.
and p is positive since parabola is concave 
upward, so

4p+=+6
p=6%2F4+=+3%2F2

The vertex is p or 3%2F2 units above the directrix.

Therefore the vertex is (-2,51%2F2) or (-2,11%2F2)

and the equation is

%28x-%28-2%29%29%5E2=4%283%2F2%29%28y-%2811%2F2%29%29

%28x%2B2%29%5E2=6%28y-11%2F2%29

That is all you wanted. But let's plot the
other parts and draw the graph.

Let's plot the vertex with an "o",

 

Now the focus is P=11%2F2 units above the
vertex, so we plot it at (-2,7)



Let's draw the latus rectum:

The latus rectum is the horizontal line segment the ends
of which are on the parabola.  Since the latus rectum is 
6 units long, we draw it 3 units right of the focus and
3 units left of the focus:


 
Now we can sketch in the parabola through the vertex and the 
end-points of the latus rectum:
 


Edwin