The line segment joining the points A(2, 3, 4 | vedclass

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Question

(A) Let the $yz$-plane intersect the line segment joining points $A(2, 3, 4)$ and $B(-3, 5, -4)$ in the ratio $\lambda : 1$ at point $M$. Using the se

Vedclass · Accepted Answer

$\left(0, \frac{19}{5}, \frac{4}{5}\right)$

Vedclass · Answer

(A) Let the $yz$-plane intersect the line segment joining points $A(2, 3, 4)$ and $B(-3, 5, -4)$ in the ratio $\lambda : 1$ at point $M$. Using the section formula,the coordinates of point $M$ are given by: $M = \left( \frac{-3\lambda + 2}{\lambda + 1}, \frac{5\lambda + 3}{\lambda + 1}, \frac{-4\lambda + 4}{\lambda + 1} \right)$. Since the point $M$ lies on the $yz$-plane,its $x$-coordinate must be $0$. Therefore,$\frac{-3\lambda + 2}{\lambda + 1} = 0$,which implies $-3\lambda + 2 = 0$,so $\lambda = \frac{2}{3}$. Substituting $\lambda = \frac{2}{3}$ into the coordinates of $M$: $y = \frac{5(2/3) + 3}{(2/3) + 1} = \frac{10/3 + 9/3}{5/3} = \frac{19}{5}$. $z = \frac{-4(2/3) + 4}{(2/3) + 1} = \frac{-8/3 + 12/3}{5/3} = \frac{4}{5}$. Thus,the point of intersection is $\left(0, \frac{19}{5}, \frac{4}{5}\right)$. Hence,option $A$ is correct.

The line segment joining the points $A(2, 3, 4)$ and $B(-3, 5, -4)$ intersects the $yz$-plane at the point:

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