Q 27. - Calculus

Question

Find the equations of the planes determined by the given conditions.

The plane contains the points $\left(x_0,0,0\right),\left(0,y_0,0\right)$ and $\left(0,0,z_0\right)$

Step-by-Step Solution

Answer

The equation of the plane that contains the points $\left(x_0,0,0\right),\left(0,y_0,0\right)$ and $\left(0,0,z_0\right)$ is $y_0{z}_{0}x+x_0{z}_{0}y+x_0{y}_{0}z=x_0{y}_0{z}_0$

Step 1: Given information

The plane that contains the points $\left(x_0,0,0\right),\left(0,y_0,0\right)$ and $\left(0,0,z_0\right)$

Step 2: Calculation

The goal is to determine the plane equation that is determined by the given conditions.

The general form of the equation of the plane is:

$a x+b y+c z=d$

The equation of the plane satisfying the points $\left(x_0,0,0\right),\left(0,y_0,0\right)$ and $\left(0,0,z_0\right)$ is obtained by substituting the points in $a x+b y+c z=d$ and solve for the constants $a, b, c$ and $d$

The point $\left(x_0,0,0\right)$ gives:

$$\begin{gathered} a\left(x_0\right)+b\left(0\right)+c\left(0\right)=d\text{ (Substitution) } \\ a{x}_0=d(\text{ Simplify) } \\ a=\frac{d}{x_0} \end{gathered}$$

The point $\left(0,y_0,0\right)$ gives:

$a\left(0\right)+b\left(y_0\right)+c\left(0\right)=d$ (Substitution)

$b=\frac{d}{y_0}\text{ (Simplify) }$

The point $\left(0,0,z_0\right)$ gives:

$$\begin{gathered} a\left(0\right)+b\left(0\right)+c\left(z_0\right)=d\text{ (Substitution) } \\ c=\frac{d}{z_0}(\text{ Simplify) } \end{gathered}$$

Step 3: Calculation

In the general form, substitute the values of the constants $a, b$ and $c$

The equation of the plane is:

$\frac{d}{x_0}x+\frac{d}{y_0}y+\frac{d}{z_0}z=d$ (Substitution)

$d\left(\frac{1}{x_0}x+\frac{1}{y_0}y+\frac{1}{z_0}z\right)=d$ (Take common factor out)

$\frac{x}{x_0}+\frac{y}{y_0}+\frac{z}{z_0}=1$

$\frac{y_0{z}_{0}x+x_0{z}_{0}y+x_0{y}_{0}z}{x_0{y}_0{z}_0}=1$

$y_0{z}_{0}x+x_0{z}_{0}y+x_0{y}_{0}z=x_0{y}_0{z}_0$

Thus, the equation of the plane that contains the points $\left(x_0,0,0\right),\left(0,y_0,0\right)$ and $\left(0,0,z_0\right)$ is $y_0{z}_{0}x+x_0{z}_{0}y+x_0{y}_{0}z=x_0{y}_0{z}_0$