What is the standard form of the equation of the parabola with a focus at (-1,7) and a directrix of #y= 3#?

1 Answer
Jim G.
Feb 6, 2018

#(x+1)^2=8(y-5)#

Explanation:

#"for any point "(x,y)" on the parabola"#

#"the distance to the focus and directrix are equal"#

#"using the "color(blue)"distance formula"#

#•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

#"let "(x_1,y_1)=(-1,7)" and "(x_2,y_2)=(x,y)#

#d=sqrt((x+1)^2+(y-7)^2)=|y-3|#

#color(blue)"square both sides"#

#(x+1)^2+(y-7)^2=(y-3)^2#

#rArr(x+1)^2=(y-3)^2-(y-7)^2#

#color(white)((x+1)^2xxx)=cancel(y^2)-6y+9cancel(-y^2)+14y-49#

#color(white)(xxxxxxxx)=8y-40#

#rArr(x+1)^2=8(y-5)#

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