The number of pairs (x,y) which will satisfy the equation `x^2-x y+y^2=4(x+y-4)` is

Options

A. 1
B. 2
C. 4
D. none of these

Answer

A

Explanation

We have `x^(2)-xy+y^(2)-4(x-y-4)`
`impliesx^(2)-x(y+4)+y^(2)-4y+16=0`
`:' x epsilonR`
`:.(-(y+4))^(2)-4.1.(y^(2)-4y+16)ge0` [using `b^(2)-4acge0`]
`impliesy^(2)+8y+16-4y^(2)+16y-64ge0`
`implies3y^(2)-24y+48le0`
`impliesy^(2)-8y+16le0implies(y-4)^(2)le0`
`:.(y-4)^(2)=0`
`:.y=4`
Then `x^(2)-4x+16=4(x+4-4)`
`x^(2)-8x+16=0`
`(x-4)^(2)=0`
`x=4`
Number of pairs is 1 i.e. `(4,4)`