Question

The coordinates of the focus of the parabola y² − x − 2y + 2 = 0 are
(a) (5/4, 1)
(b) (1/4, 0)
(c) (1, 1)
(d) none of these

Answer 1

(a) (5/4, 1)

Given:
The equation of the parabola is y² − x − 2y + 2 = 0.
$$\begin{gathered} \Rightarrow {\left(y-1\right)}^2-1=\left(x-2\right) \\ \Rightarrow {\left(y-1\right)}^2=x-1 \end{gathered}$$

Let $X=x-1,Y=y-1$

∴ $Y^2=X$

Comparing with $Y^2=4aX$:
$a=\frac{1}{4}$

Focus = $\left(X=a,Y=0\right)=\left(X=\frac{1}{4},Y=0\right)=\left(x=\frac{1}{4}+1,y=1\right)=\left(x=\frac{5}{4},y=1\right)$

Hence, the focus is at (5/4, 1).