Question

Write the ratio in which the plane 4x + 5y − 3z = 8 divides the line segment joining the points (−2, 1, 5) and (3, 3, 2).

Answer 1

$$\begin{gathered} \text{We know that the ratio in which the plane}ax+by+cz+d=0\text{divides the line segment joining}\left(x_1,y_1,z_1\right)\text{and}\left(x_2,y_2,z_2\right)\text{is} \\ \frac{-\left(a{x}_1+b{y}_1+c{z}_1+d\right)}{a{x}_2+b{y}_2+c{z}_2+d} \\ \text{Here,}a=4;b=5;c=-3;d=-8;x_1=-2;y_1=1;z_1=5;x_2=3;y_2=3;z_2=2 \\ \text{So, the required ratio} \\ =\frac{-\left(4\left(-2\right)+5\left(1\right)-3\left(5\right)-8\right)}{4\left(3\right)+5\left(3\right)-3\left(2\right)-8} \\ =\frac{-\left(-8+5-15-8\right)}{12+15-6-8} \\ =\frac{26}{13} \\ =\frac{2}{1}\text{or}2:1 \end{gathered}$$