Please solve below problem and clearly state the answers Fin

Please solve below problem and clearly state the answers.

Find the vertices and foci of the ellipse. 4x^2 + y^2 = -2y vertices (x, y) = () (smaller y-value) (x, y) = () (larger y-value) foci (x, Y) = () (smaller y-value) (x, Y) = () (x, Y) = (larger y-value)

Solution

4x²+ y²=-2y

4x²+ y²+2y+1=1

4x²+ (y+1)²=1

x²/(1/4) + (y+1)²/1=1

This is of the form (x-h)²/b² + (y-k)²/a²=1

Where h=0 and k =-1

a=1 b=1/2

c²=a²-b²

c²= 1- 1/4

c=sqrt3/2

vertices= (h,k+a),(h,k-a) = (0,-1 + 1),(0,-1-1)= (0,0),(0,-2)

smaller value=(0,-2)

larger value=(0,0)

focii=(h,k+c),(h,k-c)= (0,-1 + sqrt3/2),(0,-1-sqrt3/2)

smaller value= (0,-1-sqrt3/2)

larger value=(0,-1+sqrt3/2)