Question

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

(a) $(0,0)$

(b) $\left(\frac{1}{2}, \frac{1}{4}\right)$

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

(d) None of these

Answer 1

$\because y=2\left(x^2+\frac{x}{2}+\frac{1}{16}\right)-\frac{1}{8} \\$

$\Rightarrow  \left(x+\frac{1}{4}\right)^2=\frac{1}{2}\left(y+\frac{1}{8}\right) \\$

$\Rightarrow  X^2=4 A Y$

$ \text { where, }  X=x+\frac{1}{4}, Y=y+\frac{1}{8} \text { and } A=\frac{1}{8} \\$

$\therefore  \text { Focus }=(0, A) \\$

$\therefore  x+\frac{1}{4}=0, y+\frac{1}{8}=\frac{1}{8} \\$

$\Rightarrow  x=-\frac{1}{4}, y=0$

So, The correct option of this question will be (C).