A parabola is the curve formed from all the points (x, y) that are equidistant f

Question

A parabola is the curve formed from all the points (x, y) that are equidistant from the directrix (a line) and the focus (a point). Let's say the focus is at point F(s, t)* and the directrix is the line is y = D (Note that, in general, the directrix doesn't need to be a horizontal line). (a) Briefly explain why the equation squareroot (x - s)^2 + (y - t)^2 = |y - D| aligns with the definition of parabola given above. (b) It is convenient to replace |y - D| with squareroot (y - D)^2 in the equation above. Briefly explain why that is permissible (one sentence is fine). (c) Rewrite squareroot (x - s)^2 + (y - t)^2 = squareroot (y - D)^2 so it has the same structure as y = a(x - h)^2 + k. (d) Use your answer from part (c) to find the focus and directrix of y = x^2 - 3x + 2

Explanation / Answer

Focus is F ( s ,t) and let P(x,y) be any point on the parabola

the distance PF= [ (x-s)2+(y-t)2]1/2

directrix is y=d the distance of P from y=D is

PQ = |y-D | where Q is the foot of the perpendicular from P to the line y=D

   Definition of parabola is PF = PQ ---(1)

   b . | y-D| = positive square root of (y-D)2 =+[ ( y-D)2 ]1/2

c . squaring (1) on either side the equality [ (x-s)2+(y-t)2]1/2=[ ( y-D)2 ]1/2

[ (x-s)2+(y-t)2]=[ ( y-D)2 ]  

simplifying

we gt x2 - 2xs+s2 - ay + b=0 => (x-s)2 =ay -b which represents parabola

d. y= x2-3x+2 = (x-3/2)2- 1/4

   y + 1/4 = (x-3/2)2

is of the form Y = X2  where Y = y+1/4 and X= x-3/2

for   Y = X2 , 4a=1 and a= 1/4    vertex = (0,0) and focus = ( 0,a) and directrix Y = -a

Vertex : Y= y+1/ =0 ie y= -1/4 , X= x-3/2 =0 ie x= 3/2 hence vertex= ( 3/2 , 1/4 )

Focus X= 0 ie x= 3/2 , Y= a= 1/4 ie y+1/4 =1/4 ie y=0 focus ( 3/2 ,0)

Directrix Y = - a ie y+1/4 = -1/4 hence the directrix is y= -1/2

  

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