Question

The coordinates of the focus of the parabola y² − x − 2y + 2 = 0 are 

Options

•  (5/4, 1) 

• (1/4, 0) 

• (1, 1) 

•  none of these 

Answer 1

(5/4, 1)
Given:
The equation of the parabola is y² − x − 2y + 2 = 0.

\[\Rightarrow \left( y - 1 \right)^2 - 1 = \left( x - 2 \right)\]
\[ \Rightarrow \left( y - 1 \right)^2 = x - 1\] 

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).