The sum of the coordinates of the point of contact of the line $x - 2 y = 1$ with parabola $y^{2} = 2 (x - 3)$ is

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Solution

$x = 1 + 2 y$
Solving line with parabola $y^{2} = 2 (x - 3),$ we have
$y^{2} = 2 (1 + 2 y - 3) \Rightarrow y^{2} = 2 (2 y - 2) \Rightarrow y^{2} - 4 y + 4 = 0 \Rightarrow y = 2 \Rightarrow x = 5$
Hence required sum is $5 + 2 = 7$

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