What is the standard form of the equation of the parabola with a focus at (5,-3) and a directrix of #y= -2#?

1 Answer
Jim G.
Feb 12, 2018

#y=-1/2x^2+5x-15#

Explanation:

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

#"from "(x,y)" to the focus/directrix are equal"#

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

#sqrt((x-5)^2+(y+3)^2)=|y+2|#

#color(blue)"squaring both sides"#

#(x-5)^2+(y+3)^2=(y+2)^2#

#rArr(y+3)^2-(y+2)^2=-(x-5)^2#

#rArrcancel(y^2)+6y+9cancel(-y^2)-4y-4=-x^2+10x-25#

#rArr2y=-x^2+10x-30#

#rArry=-1/2x^2+5x-15larrcolor(red)"in standard form"#

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