Derive the equation of the parabola with focus: (-6, -3/2) and directrix: y = 11/2 explain, i need !

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thhe geometric definition of a parabola:

a set of points where each point has its distance from a fixed point equal to its distance for a fixed line. that fixed point is called the focus and that fixed line is called the directrix.

in other words, the points on a parabola are equidistant from a focus point and a directrix line.

the key to deriving this is using the distance formula.
the distance formula gives the distance between two points:   and  . the distance is

     

if we have a point  on this parabola, the distance between  and the focus point is 

   

now, the distance between that  same point  and the directrix line at  . when talking about distances from a point to a line, we want the shortest possible distance.

the directrix is a horizontal line, meaning that all the points on the line have the same y-coordinate. the points on the horizontal line can have any x-coordinate. 

if we choose an x-coordinate that is the same as the x-coord for  , then there is a point on the directrix  . this will get us the shortest distance.

so the distance between on the parabola and the point  on the directrix line (they share the same x-coordinate)

   

the definition of a parabola says that these two distances are equal.

   

now we can simplify to get a standard form equation of a parabola, which is  . the amount of simplification depends on your teacher. 

get rid of the square roots by squaring both sides of the equation.

   

i will do all expansions separately.
   
there is are formulas to with expanding binomials that are squared:

   
   

if we expand  then

   

if we expand  then

   

if we expand  then

   

going back to our original equation, let us put in all the expansions involving y first.

   

combine like terms. i first subtract y² from both sides, then subtract 9/4 from both sides.

   

expanding out  (also, 112/4 = 28)

   

so your parabola has the equation 

   

 

if you want vertex form

   

then do not expand out  . leave it alone at this step (i also simplify 112/4 to 28)

   

solve for y to get vertex form

   

so vertex form version is