91Ó°ÊÓ

The given point is the $\left(\mathrm{Î\pm },\mathrm{β},\mathrm{γ}\right)$and the constraint is$ax+by+cz=d.$

It is given that the point is $\left(\mathrm{Î\pm },\mathrm{β},\mathrm{γ}\right),$so the distance of a point from the given point is $\sqrt{(x−\mathrm{Î\pm }{)}^2+(y−\mathrm{β}{)}^2+(z−\mathrm{γ}{)}^2}.$

Let the function be the square of the above distance $D(x,y,z)=(x−\mathrm{Î\pm }{)}^2+(y−\mathrm{β}{)}^2+(z−\mathrm{γ}{)}^2$and the constraint as$g(x,y,z)=ax+by+cz−d.$

By using the Lagrange's method, the gradient of the above functions are,

$$\begin{gathered} \mathrm{∇}D(x,y,z)=2(x−\mathrm{Î\pm })\mathbf{i}+2(y−\mathrm{β})\mathbf{j}+2(z−\mathrm{γ})\mathit{k} \\ \mathrm{∇}\mathrm{g}(\mathrm{x},\mathrm{y},\mathrm{z})=\mathrm{a}\mathbf{i}+\mathrm{b}\mathbf{j}+\mathrm{c}\mathbf{k} \end{gathered}$$

Now, the system of the equation is

$$\begin{gathered} \mathrm{∇}D(x,y,z)=\mathrm{λ}\mathrm{∇}g(x,y,z) \\ 2(x−\mathrm{Î\pm })\mathbf{i}+2(y−\mathrm{β})\mathbf{j}+2(z−\mathrm{γ})\mathbf{k}=\mathrm{λ}(a\mathbf{i}+b\mathbf{j}+c\mathbf{k}) \end{gathered}$$

Here,$2(x−\mathrm{Î\pm })=a\mathrm{λ},2(y−\mathrm{β})=b\mathrm{λ}\text{and}2(z−\mathrm{γ})=c\mathrm{λ}.$

The values of $\mathrm{λ},$we get are$\frac{2(x−\mathrm{Î\pm })}{a}=\frac{2(y−\mathrm{β})}{b}=\frac{2(z−\mathrm{γ})}{c}=\mathrm{λ}.$

So, $y=\mathrm{β}+\frac{b}{a}(x−\mathrm{Î\pm })\text{and}z=\mathrm{γ}+\frac{c}{a}(x−\mathrm{Î\pm }).$

Substitute the above values in the given constraint,

$$\begin{gathered} ax+by+cz=d \\ ax+b(\mathrm{β}+\frac{b}{a}(x−\mathrm{Î\pm }\left)\right)+c(\mathrm{γ}+\frac{c}{a}(x−\mathrm{Î\pm }\left)\right)=d \\ (x−\mathrm{Î\pm })(a^2+b^2+c^2)=ad−a^2\mathrm{Î\pm }−ab\mathrm{β}−ac\mathrm{γ} \\ x−\mathrm{Î\pm }=\frac{a(d−a\mathrm{Î\pm }−b\mathrm{β}−c\mathrm{γ})}{a^2+b^2+c^2} \\ x=\mathrm{Î\pm }+\frac{a(d−a\mathrm{Î\pm }−b\mathrm{β}−c\mathrm{γ})}{a^2+b^2+c^2} \end{gathered}$$

Thus, the points where the extreme value may appear are

role="math" localid="1649940586106" $\left(\mathrm{Î\pm }+\frac{\mathrm{a}(\mathrm{d}−\mathrm{²¹Î\pm }−\mathrm{²úβ}−\mathrm{³¦Î³})}{{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2},\mathrm{β}+\frac{\mathrm{b}(\mathrm{d}−\mathrm{²¹Î\pm }−\mathrm{²úβ}−\mathrm{³¦Î³})}{{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2},\mathrm{γ}+\frac{\mathrm{c}(\mathrm{d}−\mathrm{²¹Î\pm }−\mathrm{²úβ}−\mathrm{³¦Î³})}{{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2}\right).$

Hence the points on the given surface closest to$$\begin{gathered} \left(\mathrm{Î\pm },\mathrm{β},\mathrm{γ}\right)are \\ \left(\mathrm{Î\pm }+\frac{\mathrm{a}(\mathrm{d}−\mathrm{²¹Î\pm }−\mathrm{²úβ}−\mathrm{³¦Î³})}{{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2},\mathrm{β}+\frac{\mathrm{b}(\mathrm{d}−\mathrm{²¹Î\pm }−\mathrm{²úβ}−\mathrm{³¦Î³})}{{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2},\mathrm{γ}+\frac{\mathrm{c}(\mathrm{d}−\mathrm{²¹Î\pm }−\mathrm{²úβ}−\mathrm{³¦Î³})}{{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2}\right). \end{gathered}$$