A) \[\left( \frac{1}{4},0 \right)\]
B) \[(1,2)\]
C) \[\left( \frac{5}{4},1 \right)\]
D) \[\left( \frac{3}{4},\frac{5}{2} \right)\]
Correct Answer: C
Solution :
We have the equation of parabola \[{{y}^{2}}-x-2y+2=0\] \[\Rightarrow \] \[{{y}^{2}}-2y+1=x-1\] \[\Rightarrow \] \[{{(y-1)}^{2}}=x-1\] \[\Rightarrow \] \[{{Y}^{2}}=X\] \[\Rightarrow \] \[{{Y}^{2}}=4.\left( \frac{1}{4}X \right),\] where \[Y=y-1\]and \[X=x-1\] Its focus is given by \[(a,0)=\left( \frac{1}{4},0 \right)\] ie, \[X=\frac{1}{4}\Rightarrow x-1=\frac{1}{4}\] \[\Rightarrow \] \[x=\frac{5}{4}\] and \[Y=0\] \[\Rightarrow \] \[y-1=0\] \[\Rightarrow \] \[y=1\] Hence, focus of given hyperbola is\[\left( \frac{5}{4},1 \right)\]
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