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

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(C) The given equation of the parabola is $y = 2x^2 + x$.Divide by $2$: $x^2 + \frac{x}{2} = \frac{y}{2}$.Complete the square: $(x + \frac{1}{4})^2 =

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

Vedclass · Answer

(C) The given equation of the parabola is $y = 2x^2 + x$.Divide by $2$: $x^2 + \frac{x}{2} = \frac{y}{2}$.Complete the square: $(x + \frac{1}{4})^2 = \frac{y}{2} + \frac{1}{16}$.$(x + \frac{1}{4})^2 = \frac{1}{2}(y + \frac{1}{8})$.Comparing this with the standard form $X^2 = 4AY$,where $X = x + \frac{1}{4}$,$Y = y + \frac{1}{8}$,and $4A = \frac{1}{2} \Rightarrow A = \frac{1}{8}$.The focus of the standard parabola $X^2 = 4AY$ is $(0, A) = (0, \frac{1}{8})$.Thus,$X = 0$ and $Y = \frac{1}{8}$.Substituting back: $x + \frac{1}{4} = 0 \Rightarrow x = -\frac{1}{4}$ and $y + \frac{1}{8} = \frac{1}{8} \Rightarrow y = 0$.Therefore,the focus is $(-\frac{1}{4}, 0)$.

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

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