10.2 parabola,ellipse,hyperbola — MATHS STD 11 Science — Question

$y = 2{x^2} + x$

$ \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{1}{2}\left( {y + \frac{1}{8}} \right)$

It can be written as, ${X^2} = \frac{1}{2}Y$.....$(i)$

Here $A = \frac{1}{8}$, focus of $(i)$ is $\left( {0,\frac{1}{8}} \right)$

$i.e.$ $X = 0$, $Y = \frac{1}{8}$

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

$ \Rightarrow \,x = - \frac{1}{4},$ $y = 0$

$i.e.$ focus of given parabola is $\left( { - \frac{1}{4},\,0} \right)$.