Question

write the equation of a parabola with the focus of (-2,4) and directrix y = 2 show your work and draw a sketch

Answer 1

Given the following question:

focus = (-2, 4)

directrix = y = 2

`\begin{gathered} ((ax+by+c)^2)/(a^2+b^2)=(x-f1)^2+(y-f2)^2 \\ (-2,4)=(f1,f2) \\ y=2=y-2=0=0x+1y-2=0,a=0,b=1,c=-2 \\ ((ax+by+c)^2)/(a^2+b^2)=(x-f1)^2+(y-f2)^2 \\ (0x+1y-2)^2)/(0^2+1^2)=(x-(-2))^2_{}+(y-4)^2 \\ ((y-2)^2)/(1)=(x+2)^2+(y-4)^2 \\ (y-2)^2=(x+2)^2+(y-4)^2 \\ y^2-4y+4=(x^2+4x+4)+(y^2-8y+16) \\ y^2-4y+4=x^2+4x+4+y^2-8y+16 \\ -4y=x^2+4x-8y+16+8 \\ -4y+8y=x^2+4x+16 \\ 4y=x^2+4x+16\colon4 \\ 4y=x^2+4x+16\colon4=y=(1)/(4)x^2+x+4 \\ y=(1)/(4)x^2+x+4 \end{gathered}`

Now for the sketch: