Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

The Cartesian equation of the given plane is $x-2y+3z=17$.

Let the plane divide the line segment joining the points $A(-1,4,7)$ and $B(3,-5,8)$ in the ratio $\lambda :1$.

The coordinates of the point of intersection are given by section formula as $\left(\frac{3\lambda -1}{\lambda +1},\frac{-5\lambda +4}{\lambda +1},\frac{8\lambda +7}{\lambda +1}\right)$.

Since this point lies on the plane $x-2y+3z=17$, we substitute the coordinates into the equation of the plane:

$\frac{3\lambda -1}{\lambda +1}-2\left(\frac{-5\lambda +4}{\lambda +1}\right)+3\left(\frac{8\lambda +7}{\lambda +1}\right)=17$

$3\lambda -1+10\lambda -8+24\lambda +21=17(\lambda +1)$

$37\lambda +12=17\lambda +17$

$20\lambda =5$

$\lambda =\frac{1}{4}$

The required ratio is $1:4$.

Answer: $1:4$