Find the points on the surface y2 = 25 + xz that are closest to the origin.

You are essentially minimizing [tex]x^2+y^2+z^2[/tex] subject to [tex]y^2=25+xz[/tex]. (The distance between the origin and any point [tex](x,y,z)[/tex] on the given surface is [tex]\sqrt{x^2+y^2+z^2}[/tex], but [tex]\sqrt{\mathrm{func}}[/tex] and [tex]\mathrm{func}[/tex] share the same critical points.)

Via Lagrange multipliers, we have Lagrangian

[tex]L(x,y,z,\lambda)=x^2+y^2+z^2+\lambda(y^2-xz-25)[/tex]

with partial derivatives (set equal to 0)

[tex]L_x=2x-\lambda z=0\implies 2x=\lambda z[/tex]
[tex]L_y=2y+2\lambda y=0\implies y(1+\lambda)=0[/tex]
[tex]L_z=2z-\lambda x=0\implies 2z=\lambda x[/tex]
[tex]L_\lambda=y^2-xz-25=0\implies y^2=xz+25[/tex]

[tex]L_x=0\implies zL_x=0\implies 2xz=\lambda z^2[/tex]
[tex]L_z=0\implies xL_z=0\implies 2xz=\lambda x^2[/tex]
[tex]zL_x-xL_z=\lambda(z^2-x^2)=0[/tex]

We assume [tex]\lambda\neq0[/tex], which means [tex]z^2=x^2\implies |z|=|x|[/tex].

[tex]L_y=0\implies y(1+\lambda)=0\implies y=0\text{ or }\lambda=-1[/tex]

In the first case, we have

[tex]y^2=25+xz\implies -xz=25[/tex]

which means one of [tex]x,z[/tex] must be positive, and the other is negative. From [tex]|x|=|z|[/tex] we have [tex]x=-z[/tex], so

[tex]-(-z)z=z^2=25\implies z=\pm5\implies x=\mp5[/tex]

So we get two critical points, (-5, 0, 5) and (5, 0, -5).

In the second case, if [tex]\lambda=-1[/tex], we get

[tex]\begin{cases}2x=-z\\2z=-x\end{cases}\implies x=z=0[/tex]

which leads us to

[tex]y^2=25\implies y=\pm5[/tex]

i.e. we have two additional critical points (0, 5, 0) and (0, -5, 0).

At each of these points, we get respective distances from the origin of [tex]\{5\sqrt2,5\sqrt2,5,5\}[/tex], so the two closest points to the origin on the surface [tex]y^2=25+xz[/tex] are (0, 5, 0) and (0, -5, 0).