How do you find the vertex, the focus, and the directrix of the parabola #x^2 -8x -28y -124=0#?
1 Answer
Aug 14, 2017
Explanation:
#"the translated forms of the equation of a parabola are"#
#•color(white)(x)(y-k)^2=4p(x-h)larrcolor(blue)" opens horizontally"#
#•color(white)(x)(x-h)^2=4p(y-k)larrcolor(blue)" opens vertically"#
#"in both cases the "color(magenta)"vertex "=(h,k)#
#"p is the distance from the vertex to the focus "#
#"and directrix"#
#color(blue)"completing the square"" on "x^2-8x#
#"and moving all other terms to the right"#
#x^2-8xcolor(red)(+16)=28y+124color(red)(+16)#
#(x-4)^2=28y+140#
#rArr(x-4)^2=28(y+5)#
#rArrcolor(magenta)"vertex "=(4,-5)#
#"the parabola opens vertically up "4p>0#
#4p=28rArrp=7#
#rArrcolor(red)"focus "=(4,-5+7)=(4,2)larr" above the vertex"#
#color(blue)"directrix is "y=(-5-7)=-12larr" below the vertex"#
