Derive the equation of the parabola with a focus at (0, –4) and a directrix of y = 4. (2 points)A f(x) = –16x2B f(x) = – x2C f(x) = x2D f(x) = 16x2

Accepted Solution

Equation of parabola with focus at (0,-4) and directrix is y=4 .As we know parabola is the locus of all the points such that distance from fixed point on the parabola to fixed line directrix is same.The parabola is opening downwards.Let any point on parabola is (x,y).Distance from focus(0,-4) to (x,y) = [tex]\sqrt{(x-0)^{2} +(y+4) ^{2}}=\sqrt{x^{2}+ (y+4)^{2}}[/tex]Distance from (x,y) to directrix, y=4 is =[tex]\left | y-4 \right |[/tex]As these distances are equal.[tex]\sqrt{x^{2}+ (y+4)^{2}}=\left | y-4 \right |\\{x^{2}+ (y+4)^{2}=(y-4)^{2}[/tex]→x²+y²+8 y +16 = y² - 8 y+16→ x² = -8 y - 8 y= -16 y   [ Cancelling y² and 16 from L.H.S and R.H.S ]So , equation of parabola is , x²= - 16 y or f(x)= -x²/16