The focus of the parabola y = 2x^{2} + x is | vedclass

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(C) The given equation of the parabola is $y = 2x^{2} + x$. Dividing by $2$,we get $x^{2} + \frac{x}{2} = \frac{y}{2}$. Completing the square,$x^{2} +

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$(-\frac{1}{4}, 0)$

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(C) The given equation of the parabola is $y = 2x^{2} + x$. Dividing by $2$,we get $x^{2} + \frac{x}{2} = \frac{y}{2}$. Completing the square,$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{y}{2} + \frac{1}{16}$. This simplifies to $(x + \frac{1}{4})^{2} = \frac{1}{2}(y + \frac{1}{8})$. Comparing this with the standard form $X^{2} = 4AY$,where $X = x + \frac{1}{4}$,$Y = y + \frac{1}{8}$,and $4A = \frac{1}{2}$,we find $A = \frac{1}{8}$. The focus in the $(X, Y)$ coordinate system is $(0, A) = (0, \frac{1}{8})$. Substituting back,$x + \frac{1}{4} = 0 \Rightarrow x = -\frac{1}{4}$ and $y + \frac{1}{8} = \frac{1}{8} \Rightarrow y = 0$. Thus,the focus of the given parabola is $(-\frac{1}{4}, 0)$.

The focus of the parabola $y = 2x^{2} + x$ is

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