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Condition for Concurrency of Three Lines

The line x-...

Question

The line $x - 3 y + 7 = 0$ intersects the pair of straight lines $x^{2} + 2 y^{2} - 3 x y + 2 x - 5 y + 3 = 0$ in two points $A$ and $B$ . The midpoint of $A B$ is $(p, q)$ , then find the value of $p - q .$

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Solution

${(3 y - 7)}^{2} + 2 y^{2} - 3 (3 y - 7) y + 2 (3 y - 7) - 5 y + 3 = 0$
$\Rightarrow 2 y^{2} - 20 y + 38 = 0$
$\Rightarrow y^{2} - 10 y + 19 = 0$
$\frac{y_{1} + y_{2}}{2} = \frac{10}{2} = 5, x = 8$
$\Rightarrow p - q = 8 - 5 = 3$

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