QJEEM24Ma442b: JEE Main 2024 Mathematics - Class 12 JEEM24Ma442b Solution: Let the point, on the line passing through the points and ,…

Line through $P Q$

$\frac{x-1}{4}=\frac{y+2}{-2}=\frac{z-3}{4}$

Any point on $P Q$ . be $R(4 \lambda+1,-2 \lambda-2,4 \lambda+3)$

$(P R)^2=81$

$(4 \lambda+1-1)^2+(-2 \lambda-2+2)^2+(4 \lambda+3-3)^2=81$

$16 \lambda^2+4 \lambda^2+16 \lambda^2=81$

$36 \lambda^2=81$

$\lambda= \pm \frac{9}{6}= \pm \frac{3}{2}$

$\therefore R$ can be $(7,-5,9)$ or $(-5,1,-3)$

Distance from origin for both points be $\sqrt{49+25+81}$ and $\sqrt{25+1+9}=\sqrt{35}$

$\therefore$ Distance of $(7,-5,9)$ is farthest from origin

$\therefore(\alpha, \beta, \gamma)=(7,-5,9)$

Now $7^2+(-5)^2+9^2=155$