1find te centerfociverticesand lengths of the major and mino

1-find te center,foci,vertices,and lengths of the major and minor axes of the graph

4x^2+25y^2+40x+200y+400=0

*the graph is an elipse

2-find the focus,vertex,and directrix of the parabola
x^2+8x+4y+16=0

Solution

4x^2 +25y^2 +40x +200y +400=0
4(x^2 +10x) +25(y^2 +8y) +400 =0
4(x^2 +10x +25) +25(y^2 +8y+16) +400 -100 -400=0
4(x+5)^2 +25(y+4)^2 = 100
(x+5)^2 / 25 +(y+4)^2/4 =1

center =(-5,-4)
major axis = 2a = 2(5) =10
minor axis =2b =2(4) =8
foci are (a+c ,b) and (a-c , b)
c^2 = 25 -9=16
c=4
foci are = (5+4 , 4) and (5-4,4)
=(9,4) and (1,4)

2).

x^2 +8x+4y +16=0

x^2 +8x +16 = -4y

(x+4)^2 = -4y

vertex is (-4,0)

directrix y =4

focus (-4,4)