Question

the angle subtended by common tangents of two ellipses 4(x-4)2 +25y2=100 4(x+1)2 + y2=4 at the origin (in degrees) is

Answer 1

equation of ellipse, 4(x - 4)² + 25y² = 100
or, (x - 4)²/5² + y²/2² = 1 , it similar of standard equation of ellipse.
and again, equation of ellipse, 4(x + 1)² + y² = 4
or, (x + 1)²/1 + y²/2² = 1

now, draw both the equation of ellipse ,
you get, a comment tangent as shown in figure . tangent touches first ellipse at (4,2) and 2nd ellipse at (-1,2) , it subtends angle at origin (0,0) .
so, slope of line joining O to 1st ellipse is 1/2
And slope of line joining O to 2nd ellipse is -2

now, angle made = tan^-1{1/2 + 2}/{1 + 1/2 × -2}
= tan^-1(∞) = π/2

hence answer is π/2