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Q.
AB is a focal chord of $x^2 - 2x + y - 2 = 0$ whose focus is S. If $AS = l_1$. then BS is equal to
Conic Sections
Solution:
Given curve is $x^2 - 2x + y - 2 = 0$
$\Rightarrow x^{2} - 2x + y - 2 +1 = 1$
$\Rightarrow \left(x -1\right)^{2} = - y + 3 = -1\left( y - 3\right)$
which is downward parabola with $ a = \frac{1}{4}$
We know, if $l_{1}$ and $l_{2}$ are the length of the segment of any focal chord then length of semi-latus rectum is
$\frac{2l_{1}l_{2}}{l_{1}+l_{2}}$
Here $AS =l_{1}$ and $BS = l_{2}$ (say) are the segments.
$\therefore \quad$ we have $\frac{2l_{1}\left(BS\right)}{l_{1}+BS} = 2a \Rightarrow BS = \frac{l_{1}}{4l_{1} - 1}$