Question

Find the vertex, focus, and directrix for the following parabolas. (a) (y - 2) = 2002 - 2) vertex : focus : directrix: (b) y2 - 4y = 20% - 22 Vertex focus : directrix (c) (z - 6) = 20(4-5) vertex focus directric (d) 22 + 402 = 4y - 8 vertex focus: directrix

Answer 1

To find the vertex,focus Eand directrix of the following parabola (y-2)* = 20(x-2) Clearly which is of the form (y = k) = 4a (x-k), a > 0 h = 2,4 = 2, 4a = 20 a=5 Vertex (V)=(h,k)= (2,2) Focus (S)=(h+a,k) = (2+5,2) =(7,2) Focus(S) = |(7,2) The equation of the directrix is x-h+a = 0 That is x-2+5=0 x+3=0 (6) 22 - 4y = 20x-22 22 - 4y = 20x-4 22-2(y) (2)+22 – 22 = 20x - 4 (y-2)* - 4 = 20x-4 (y-2)* = 20(x-0) Clearly which is of the form (y=&)* = 4a (x-h), a > 0 h = 0, k = 2, 4a = 20 =a=5 Vertex (V)= (h, k) = (0,2) Focus (S) = (h+a,k) = (0+5,2) = (5,2) Focus(S) = |(5,2) The equation of the directrix is x-h+a = 0 That is x X-0+5= 0 x+5= 0

c) (x-6)* = 20(y-5) Clearly which is of the form (x-h)* = 4a(y-k),a >0 h=6,4 = 5, 4a = 20 => a=5 Vertex (V) = (h, k) = (6,5) Focus (S) = (h, k+a) =(6,5+5) = (6,10) Focus(S)= (6,10) The equation of the directrix is y-k+a=0 That is y-5+5= 0 (y=0 (c) x2 +40x = 4y-8 x* +2x(20) +202 – 202 = 4y-8 (x+20)* = 4y-8+400 (x+20)* = 4y+392 (x+20)* = 4(y+98) Clearly which is of the form (x-h)° = 4a(y-k), a > 0 h=-20,4 = -98, 4a=4a=1 Vertex (V)= (h,k)= (-20, -98) Focus (S) = (h, k+a) =(-20, -98+1) =(-20,-97) Focus (S) = (-20,-97) The equation of the directrix is y-k+a=0 That is y +98+1=0 by +99=0