Derive the equation of the parabola with a focus at (-5, -5) and a directrix of y = 7.

Question

Derive the equation of the parabola with a focus at (−5, −5) and a directrix of y = 7.

 

A) f(x) = −one twenty-fourth (x − 1)2 − 5
B) f(x) = one twenty-fourth (x − 1)2 − 5
C) f(x) = −one twenty-fourth (x + 5)2 + 1
D) f(x) = one twenty-fourth (x + 5)2 + 1

Answer 1

you can derive the formula which becomes...

y=[1/(2(b-k))](x-a)^2+(1/2)(b+k)

(a,b)=(-5,-5) and k=7 because y=7

substitute

y=[1/(2(-5-7))](x-[-5])^2+(1/2)(-5+7)

y=[1/(-24)](x+5)^2+(1/2)(2)

y=(-1/24)(x+5)^2+1

 

you can also do the following...

the distance from the focus to a point is the same as the distance from the point to the directrix

√(y-7)^2=√(x-[-5])^2+(y-[-5])^2

square both sides and expand

y^2-14y+49=x^2+10x+25+y^2+10y+25

cancel each y^2 on both sides

-14y+49=x^2+10x+25+10y+25

-14y-10y=x^2+10x+25+25-49

-24y=(x^2+10x+25)-24

-24y=(x+5)^2-24

divide both sides by -24

y=(-1/24)(x+5)^2+1 again

 

 

Answer 2

Questions like this one require a bit of detective work. 

 

The first clues are that the directrix is horizontal and located above the focus.   Taken together, these clues point to a downward opening parabola with a vertical axis of symmetry.    Once this is noticed, it is evident that the x component of the vertex is  -5 since the vertex must be on the same vertical line of symmetry as the focus.  The vertex of a parabola is located midway between the focus and the directerix.   From this it is seen that the y component of the vertex  must be 1.   Thus the vertex is the point (-5,1).

 

Another clue is the so-called p value.   The p value is the distance between the vertex and the focus - which is also equal to the distance between the vertex and the directrix.  For this problem the p value is  7-1 = 6.  

 

A useful analytic form for a parabola is the  vertex form.  This  general form is y = a(x -x_v)² +  v_y  where

  the coordinates of the vertex are  (v_x , v_y)    .     Plugging for the now known vertex coordinates gives

 

  y =  a( x +5)²  + 1

 

The remaining unknown is the value of a.    However, a general result is that  the magnitude of a  is 1/(4 p).

 

 Since p = 6 for this problem , and the parabola opens downward (implying negative a),   a =  - 1/24

The final answer is thus:

 

   y = -(1/24) (x + 5)²  +1