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