$\begin{array}{1 1}3:2 \\ 2:3 \\ 3:4 \\ 4:3\end{array} $

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- Section formula: The coordinates of the point $C$ that divides the segment joining the points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ in the ratio $l:m$ internally is given by $C\big(\large\frac{lx_2+mx_1}{l+m},\frac{ly_2+my_1}{l+m},\frac{lz_2+mz_1}{l+m}\big)$
- The $x$ coordinate of any point on $YZ$ plane is $0$

Let ther given points be $A(-2,4,7)$ and $B(3,-5,8)$

Let the point on $YZ$ plane that divides $AB$ be $C$ and

let the ratio in which it divides be $k:1$.

$\therefore$ From section formula the coordinates of $C$ is goven by

$C\big(\large\frac{3k-2}{k+1},\frac{-5k+4}{k+1},\frac{8k+7}{k+1}\big)$

But gives that $C$ lies on $YZ$ plane.

We know that the $x$ coordinate of any point on $YZ$ plane is $0$

$\Rightarrow\:\large\frac{3k-2}{k+1}$$=0$

$\Rightarrow\;3k-2=0$ or $k=\large\frac{2}{3}$

$\therefore$ The required ratio is $\large\frac{2}{3}$$:1=2:3$

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