Question

The focus of the parabola y = 2x² + x is 

Options

• (0, 0) 

•  (1/2, 1/4) 

•  (−1/4, 0) 

• (−1/4, 1/8) 

Answer 1

(−1/4, 0) 

Given:
Equation of  the parabola = y = 2x² + x 

\[\Rightarrow x^2 + \frac{x}{2} = \frac{y}{2}\]
\[ \Rightarrow \left( x + \frac{1}{4} \right)^2 = \frac{y}{2} + \frac{1}{16}\]
\[ \Rightarrow \left( x + \frac{1}{4} \right)^2 = \frac{8y + 1}{16}\]
\[ \Rightarrow \left( x + \frac{1}{4} \right)^2 = \frac{1}{2}\left( y + \frac{1}{8} \right)\]

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).