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  4. A line, with the slope greater than one, passes through the point A (4,3) and intersects the line x-y-2=0 at the point B. If the length of the line segment AB is (√29/3), then B also lies on the line :

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Q. A line, with the slope greater than one, passes through the point $A (4,3)$ and intersects the line $x-y-2=0$ at the point $B$. If the length of the line segment $AB$ is $\frac{\sqrt{29}}{3}$, then $B$ also lies on the line :

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Solution:

Let $B\left(x_1, x_1-2\right)$
$\sqrt{\left(x_1-4\right)^2+\left(x_1-2-3\right)^2}=\frac{\sqrt{29}}{3}$
Squaring on both side $18 x _1^2-162 x _1+340=0$
$x _1=\frac{51}{9} $ or $ x _1=\frac{10}{3}$
$y _1=\frac{33}{9} $ or $ y _1=\frac{4}{3}$
Option (C) will satisfy $\left(\frac{10}{3}, \frac{4}{3}\right)$