Consider the equation 16x2+4y2+32x−16y−32=0
16(x2+2x)+4(y2−4y)=32
16(x2+2x+1−1)+4(y2−4y+4−4)=32
16(x2+2x+1)−16+4(y2−4y+4)−16=32
16(x+1)2+4(y−2)2=32+16+16
16(x+1)2+4(y−2)2=64
Dividing by 64, we get
4(x+1)2+16(y−2)2=1
Observe that the above equation is of the form
a2(x−h)2+b2(y−k)2=1
So, the given equation is ellipse
Center of the ellipse =(h,k)=(−1,2)
And a2=4,b2=16
Observe that a2<b2
To find the foci of the ellipse, find c using the relation c2=b2−a2
c2=16−4=12
⇒c=±23
Foci of the ellipse are: (h,k−c),(h,k+c)=(−1,2−23),(−1,2+23)
Ends of the major axis are: (h,k−b),(h,k+b)=(−1,2−4),(−1,2+4)=(−1,−2),(−1,6)
Ends of the minor axis are:
(h−a,k),(h+a,k)=(−1−2,2),(−1+2,2)=(−3,2),(1,2)
Ends of the latus rectum are:
(b−a2,c),(ba2,c)=(−1,23),(1,23)
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