If the equation `2x^2+4x y+7y^2-12 x-2y+t=0,w h e r et` is a parameter has exactly one real solution of hte form `(x ,y)` , then hte sum of `(x+y)` is equal to ______.

Answer

3

Explanation

`2x^(2)+4x(y-3)+7y^(2)-2y+t=0`
D=0 (for one solution)
`implies16(y-3)^(2)-8(7y^(2)-2y+t)=0`
or `2(y-3)^(2)-(7y^(2)-2y+t)=0`
or `2(y^(2)-6y+9)-(7y^(2)-2y+t)=0`
or `-5y^(2)-10y+18-t=0`
or `5y^(2)+10y+t-18=0`
Again D = 0 (for one solution)
`implies100-20(t-18)=0`
or `5-t+18=0`
or `t=23`
for `t=23,5y^(2)+10y+5=0`
`(y+1)^(2)=0ory=-1`
for `y=-1,2x^(2)-16+32=0`
`x^(2)-8x+16=0`
`x=4orx+y=3`