KVPY SA Exam 2014 Question Paper

Question : 63 of 80

Marks: +1, -0

The number of integers a in the interval [1,2014] for which the system o f equations

x + y = a, x^2/{x-1}+y^2/{ y - 1}= 4

has finitely many solutions is

Solution:

x + y = a;

x^2/{x-1}+y^2/{y-1}=4

a ∈ [1, 2014]

x + 1 + 1/{x -1}+y + 1

+ 1/{y-1}=4

(x - 1) + 1/(x-1)+ (y - 1)

+1/(y - 1)=0

(a - 2) +1/(x-1)+1/(y-1)=0

(a - 2) +(a - 2) /{(x-1) (y-1)}=0

(a - 2) [1+1/{xy+ 1- a}]=0

a ≠ 2 [for a = 2 infinitely many solutions]

xy + 1 -a + 1 = 0

x(a- x) -a + 2 = 0

⇒

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

D = a^2 + 4(2 - a) = a^2 - 4a + 8

always +ve
there two real solutions.
a ≠ 2 and a ∈ [1, 2014]
Total 2014 – 1 = 2013 values

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