SOLUTION: Find the equation of the parabola with vertex (-2,-1) & the equation of the directrix is x-2y+1=0?

The black line is the given directrix.
The black point is the given focus.
The green line is the axis of symmetry, through the vertex 
perpendicular to the directrix.
The red point is the focus, which we must find.
The point P(x,y) is an arbitrary point on the parabola.
By the definition of a parabola, the two blue lines (one from
P(x,y) to the focus and the other from P(x,y) to the directrix)
must be equal.

We find the perpendicular distance from the vertex to the directrix,
(the length oif the left blue line) by using the formula:

The perpendicular distance from the point P(x1,y1)
to the line Ax+y+C=0 is

d = 

That gives d =  or .

Then we find the focus, which is on the axis of symmetry a distance
of  from the focus (or  from the directrix).

The focus turns out to be the point ,

Next we pick an arbitrary point on the parabola P(x,y).
We set its perpendicular distance to the directrix equal to its distance
from the focus (we are setting the two blue lines equal in length):



Simplify/square both sides and after much algebra you'll end up with the
equation:



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Edwin