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NIMCET 2012

Focus of the parabola $x^2 + y^2 - 2xy - 4(x + y - 1) = 0$ is:

Solution

Given equation: 
 $x^2 + y^2 - 2xy - 4x - 4y + 4 = 0$ 

 Group terms: 
 $(x - y)^2 - 4(x + y) + 4 = 0$ 
 Let: $u = x - y,\;\; v = x + y$ 
 Then equation becomes: 
 $u^2 - 4v + 4 = 0$ 
 $u^2 = 4(v - 1)$ 
 This is the standard parabola: 
 $u^2 = 4p(v - 1)$ 
 Comparing gives: 
 $4p = 4 \Rightarrow p = 1$ 
 Vertex in $(u,v)$: $(0,1)$ 
 Focus in $(u,v)$: $(0, 1 + p) = (0,2)$ 
 Convert to $(x,y)$: 
 $x - y = 0$ 
 $x + y = 2$ 
 Solving: $x = 1,\; y = 1$ 
 Therefore, the focus is: $\boxed{(1,1)}$
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