JEE Main 2022 (Online) 25th July Morning ShiftQuestion 1
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 :
Points and Straight Lines
Solution
Let $B=(x,x-2)$. $AB^2=(x-4)^2+(x-5)^2=\frac{29}9$. $2x^2-18x+41=\frac{29}{9}\Rightarrow 18x^2-162x+340=0\Rightarrow 9x^2-81x+170=0\Rightarrow x=\frac{10}3$ or $\frac{17}3$. $B=(\tfrac{10}3,\tfrac43)$ gives slope $\frac{3-4/3}{4-10/3}=\frac52>1$; $B=(\tfrac{17}3,\tfrac{11}3)$ gives slope $\frac{3-11/3}{4-17/3}=\frac25<1$. So $B=(\tfrac{10}3,\tfrac43)$. Check $x+2y=\frac{10}3+\frac83=6$. Answer: (C).