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Q.
The focus of the parabola $y^{2} - x - 2 y + 2 = 0$ is
Conic Sections
Solution:
We have, $y^{2} - 2y = x - 2 $
or $\left(y - 1\right)^{2} = x - 1$.
Shifting the origin at $\left(1,1\right)$,
we have $x = X + 1$ , $y = Y+1$.
The equation $\left(y - 1\right)^{2} = \left(x - 1\right)$ reduces to $y^{2} = X$. This represents a parabola with latus rectum $= 1$. Coordinates of the focus w.r.t. new axes are $\left(\frac{1}{4},0\right)$. So, the coordinates of the focus w.r.t. old axes are $\left(\frac{5}{4}, 1\right)$.