Question

The number of integers a in the interval [1,2014] for which the system of equations $x+y=a;\frac{x^2}{x-1}+\frac{y^2}{y-1}=4$ has finitely many solutions is

• A. 0
• B. 1007
• C. 2013
• D. 2014

Answer 1

The correct option is C

2013

$\frac{x^2-1+1}{x+1}+\frac{y^2-1+1}{y-1}=4x+1+\frac{1}{x-1}+y+1+\frac{1}{y-1}=4a+2+\frac{1}{x-1}+\frac{1}{(a-1)-x}=4\frac{(a-1)-x+x+1}{(x-1)\left[\right(a-1)-x]}=2-a\therefore a\ne 2\text{[for a = 2 equation have infinitely many solution]}$

$\therefore$ (x - 1)[(a - 1) - x] = -1

(x - 1)[x - (a - 1)] = 1

$x^2$ - ax + (a - 2) = 0

D > 0

$\therefore$ equation have 2 real roots so

a can be 1, 3, 4....... 2014

Answer is 2013