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]