A) (0, 0)
B) \[\left( \frac{1}{2},\ \frac{1}{4} \right)\]
C) \[\left( -\frac{1}{4},\ 0 \right)\]
D) \[\left( -\frac{1}{4},\ \frac{1}{8} \right)\]
Correct Answer: C
Solution :
The given equation of parabola is \[y=2{{x}^{2}}+x\]\[\Rightarrow \,{{x}^{2}}+\frac{x}{2}=\frac{y}{2}\] \[\Rightarrow \,{{\left( x+\frac{1}{4} \right)}^{2}}=\frac{y}{2}+\frac{1}{16}\]\[\Rightarrow \,{{\left( x+\frac{1}{4} \right)}^{2}}=\frac{1}{2}\left( y+\frac{1}{8} \right)\] It can be written as, \[{{X}^{2}}=\frac{1}{2}Y\] .....(i) Here \[A=\frac{1}{8}\], focus of (i) is \[\left( 0,\frac{1}{8} \right)\] i.e. \[X=0\], \[Y=\frac{1}{8}\] Þ \[x+\frac{1}{4}=0\], \[y+\frac{1}{8}=\frac{1}{8}\]\[\Rightarrow \,x=-\frac{1}{4},\] \[y=0\] i.e. focus of given parabola is \[\left( -\frac{1}{4},\,0 \right)\].
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