Statement I : Given, equation of ellipse is ‌4x2+y2−8x−4y+4=0 ‌4x2−8x+4+y2−4y+4=4
‌⇒‌‌4(x−1)2+(y−2)2=4⇒‌
(x−1)2
1
+‌
(y−2)2
4
=1
The origin of the vertex is (1,2). Here, a2=4,b2=1 e=√1−‌
b2
a2
=√1−‌
1
4
=‌
√3
2
Equation of directrix is y−k=±‌
a
e
‌⇒y−2=±‌
2
√3∕2
⇒y−2=±‌
4√3
3
‌⇒3y−6=±4√3⇒3y=6±4√3 ∴ Statement I is true. Statement II : Given, equation of ellipse ‌x2+4y2−4x−8y+4=0 ⇒x2−4x+4+4y2−8y+4=4
⇒(x−2)2+4(y−1)2=4⇒‌
(x−2)2
4
+‌
(y−1)2
1
=1
‌∴‌
X2
4
+‌
Y2
1
=1, where X=x−2 and Y=y−1 ‌e=√1−‌
1
4
=‌
√3
2
Coordinate of focus =(±ae,0)=(±√3,0) Equation of latusrectum X‌=±√3 ⇒‌‌x−2‌=±√3⇒x=2±√3 ∴ Statement II is false.