If the line 2 x-3 y+4=0 cuts the ellipse x=3 , y=5 in A and B and ( , ) is the midpoint of AB , then 3 -2 =

Step-by-step solution

Given line 2 x-3 y+4=0 cuts the ellipse x=3 , y=5 . Here, a=3, b=5 . Required equation of ellipse is aligned & x^2 a^2 + y^2 b^2 =1 & x^2 9 + y^2 25 =1 . . aligned Satisfy the line 3 y=2 x+4 y = 2 x +4 3 From (i) aligned & 25 x ^2+9 y ^2=225 & 25 x ^2+9 ( 2 x +4 3 )^2=225 & 25 x ^2+ 9 9 (4 x ^2+16+16 x )=225 & 29 x ^2+16 x -209=0 aligned Let A is ( x ₁, y ₁ ) and B ( x ₂, y ₂ ) aligned & x ₁+ x ₂= -16 29 & x ₁+ x ₂ 2 = -8 29 aligned Here, ( , ) is the mid point of the line AB = -8 29 Now, 2 x=3 y-4 x= 3 y-4 2 From