Question

The focus of the parabola y = 2x² + x is
(a) (0, 0)
(b) (1/2, 1/4)
(c) (−1/4, 0)
(d) (−1/4, 1/8)

Answer 1

(c) (−1/4, 0)

Given:
Equation of the parabola = y = 2x² + x
$$\begin{gathered} \Rightarrow x^2+\frac{x}{2}=\frac{y}{2} \\ \Rightarrow {\left(x+\frac{1}{4}\right)}^2=\frac{y}{2}+\frac{1}{16} \\ \Rightarrow {\left(x+\frac{1}{4}\right)}^2=\frac{8y+1}{16} \\ \Rightarrow {\left(x+\frac{1}{4}\right)}^2=\frac{1}{2}\left(y+\frac{1}{8}\right) \end{gathered}$$

Let $X=x+\frac{1}{4},Y=y+\frac{1}{8}$
∴ $X^2=\frac{1}{2}Y$

Comparing with $X^2=4aY$:
$a=\frac{1}{8}$

Focus = $\left(X=0,Y=a\right)=\left(x=\frac{-1}{4},y=0\right)$

Hence, the focus is at (−1/4, 0).