What is the equation in standard form of the parabola with a focus at (1,4) and a directrix of y= 3?

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Answer 1 · 2018-06-01T11:53:54

The equation of parabola is #y=1/2(x-1)^2+3.5 #

Focus is at #(1,4) #and directrix is #y=3#. Vertex is at midway

between focus and directrix. Therefore vertex is at #(1,(4+3)/2)#

or at #(1,3.5)# . The vertex form of equation of parabola is

#y=a(x-h)^2+k ; (h.k) ;# being vertex. # h=1 and k = 3.5#

So the equation of parabola is #y=a(x-1)^2+3.5 #. Distance of

vertex from directrix is #d= 3.5-3=0.5#, we know # d = 1/(4|a|)#

#:. 0.5 = 1/(4|a|) or |a|= 1/(0.5*4)=1/2#. Here the directrix is

below the vertex , so parabola opens upward and #a# is positive.

#:. a=1/2# . The equation of parabola is #y=1/2(x-1)^2+3.5 #

graph{0.5(x-1)^2+3.5 [-20, 20, -10, 10]} [Ans]