Geometric Applications

$z = x y,(2 ; 1 ; 2)$;

$z = x^{2}+y^{2},(1 ; 1 ; 2)$;

$z = 2 x^{2}-4 y^{2},(-2 ; 1 ; 4)$;

$z = (x-y)^{2}-x+2 y,(1 ; 1 ; 1)$;

$z = x^{3}-3 x y+y^{3},(1 ; 1 ;-1)$;

$z = \sqrt{x^{2}+y^{2}}-x y,(-3 ; 4 ; 17)$;

$z = \sqrt{x^{2}+y^{4}},(0 ; 0 ; 0)$;

$z = x-y+\sqrt{\left|x y\right|},(0 ; 0 ; 0)$;

$z = \ln \sqrt{x^{2}+y^{2}},(0 ; 1 ; 0)$;

$z = \sin (x / y),(\pi ; 1 ; 0)$;

$z = e^{x \cos y},(1 ; 0 ; e)$;

$z = \operatorname {arctg}(y / x),(1 ; 1 ; \pi / 4)$.

$x^{2}+y^{2}+z^{2} = 169,(3 ; 4 ;-12)$;

$x y^{2}+z^{3} = 12,(1 ; 2 ; 2)$

$x^{3}+y^{3}+z^{3}+x y z = 6,(1 ; 2 ;-1)$;

$x y z\left(z^{2}-x^{2}\right) = 6+y^{5},(1 ; 1 ; 2)$;

$\sqrt{x^{2}+y^{2}+z^{2}} = x+y+z-4,(2 ; 3 ; 6)$;

$e^{z}-z+x y = 3$, $(2 ; 1 ; 0)$;

$z = y+\ln (x / z),(1 ; 1 ; 1)$;

$2^{x / z}+2^{y / z} = 8,(2 ; 2 ; 1)$.

$x = u+v, y = u^{2}+v^{2}, z = u^{3}+v^{3},(3 ; 5 ; 9)$;

$x = u, y = u^{2}-2 u v, z = u^{3}-3 u^{2} v,(1 ; 3 ; 4)$;

$x = u+\ln v, y = v-\ln u, z = 2 u+v,(1 ; 1 ; 3)$;

$x = \cos u \operatorname {ch} v, y = \sin u \operatorname {ch} v, z = \operatorname {sh} v,(\operatorname {ch} 1 / \sqrt{2} ; \operatorname {ch} 1 / \sqrt{2} ; \operatorname {sh} 1)$.

$x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2} = 1$ (ellipsoid);

$x^{2} / a^{2}+y^{2} / b^{2}-z^{2} / c^{2} = -1$ (two-sheeted hyperboloid);

$x^{2} / p-y^{2} / q = 2 z$ (hyperbolic paraboloid).

$x^{2} / a^{2}+y^{2} / b^{2}-z^{2} / c^{2} = 1$ (one-sheeted hyperboloid);

$x^{2} / p+y^{2} / q = 2 z$ (elliptic paraboloid);

$x^{2} / a^{2}+y^{2} / b^{2}-z^{2} / c^{2} = 0$ (cone).

$x^{n}+y^{n}+z^{n} = a^{n}, n \in N, a > 0$

$\left(x^{2}+y^{2}+z^{2}\right)^{2} = a^{2}\left(x^{2}-y^{2}+z^{2}\right), a \neq 0$.

$x = u \cos v, y = u \sin v, z = u$;

$x = 3 \cos u \cos v, y = 2 \cos u \sin v, z = \sin u$, at the point with parameters $u = u_{0}, v = v_{0}$. Express the coefficients of the resulting equation in terms of the coordinates $x_{0}, y_{0}, z_{0}$ of the point of tangency.

$x = (b+a \cos \psi ) \cos \varphi , y = (b+a \cos \psi ) \sin \varphi , z = a \sin \psi , b \geqslant a > 0$

$x = \sin \varphi \cos \psi , y = \sin \varphi \sin \psi , z = \ln \operatorname {tg}(\varphi / 2)+\cos \varphi$ at the point with parameters $\varphi = \varphi_{0}, \psi = \psi_{0}$.

$x^{2}+y^{2}+z^{2}-6 y+4 z = 12$;

$x^{2}+y^{2}-z^{2}-2 x = 0$;

$x^{2}+2 y^{2}+3 z^{2}+2 x y+2 x z+4 y z = 8$.

$x^{2}+2 y^{2}+z^{2} = 1, x-y+2 z = 0$

$z^{2}+x y+x z = 1, x-y+2 z = 1$;

$4 x^{2}+6 y^{2}+4 z^{2}+4 x z-8 y-4 z+3 = 0, x+2 y = 0$.

$x^{2}-y^{2} = 3 z, M(0 ; 0 ;-1), x = 2 y = z$

$90 x^{2}+160 y^{2}+576 z^{2} = 2880, M(12 ;-3 ;-1), x = 0, y = 0$.

$x^{2}+y^{2}+z^{2} = 2 x,\left\{ \begin{aligned} x-y-z & = 2 \\ 2 x-2 y-z & = 4\end{aligned}\right\}$

$z = x y, x = y = -2 z$.

$z^{2} = x y, x^{2}+y^{2}+z^{2} = 1$;

$x y = a z, x^{2}+y^{2} = b^{2}, b > 0$;

$x y = a z, \sqrt{x^{2}+z^{2}}+\sqrt{y^{2}+z^{2}} = b, b > 0$.

$x^{2}+y^{2}+z^{2} = 2 a x, x^{2}+y^{2}+z^{2} = 2 b y, x^{2}+y^{2}+z^{2} = 2 c z, a \neq 0$, $b \neq 0, c \neq 0$

$x^{2}+y^{2}+z^{2} = a^{2}, y = b x, x^{2}+y^{2} = c^{2} z^{2}, a > 0, c > 0$;

$x^{2}+y^{2}+z^{2} = a^{2}, x y = b z^{2}, 2 x^{2}+z^{2} = c\left(2 y^{2}+z^{2}\right), a > 0, c > 0$.

the sum of the lengths of the segments it cuts off on the coordinate axes is smallest;

the centroid of the triangle cut out on it by the coordinate planes is at the smallest distance from the center of the ellipsoid;

the tetrahedron bounded by it and the coordinate planes has the smallest volume.

$x^{3}+y^{3}-3 x y = 0$;

$x^{3}-2 x^{2} y-y^{2} = 0$;

$x^{3}+y^{3}-3 x^{2} = 0$;

$x^{3}-2 x^{2}+x-y^{2} = 0$;

$x^{3}-2 x^{2} y+2 x y^{2}+2 x^{2}-2 x y+x = 0$;

$x^{4}+x y-y^{4} = 0$;

$y^{4}+x^{2} y^{2}-4 x^{2} = 0$;

$x^{4}+y^{4}-x^{2}-y^{2} = 0$;

$x^{2} y^{2}-(y+1)^{2}\left(4-y^{2}\right) = 0$;

$x^{4}+y^{4}-8 x^{2}-10 y^{2}+16 = 0$;

$x^{5}-\left(y-2 x^{2}\right)^{2} = 0$;

$y^{5}+5 y^{4}-4 x^{2} = 0$;

$x^{6}-(y-2)^{4}-x^{2} = 0$;

$y^{6}-y^{4}+x^{2} = 0$.

$y^{2} = a x^{2}+x^{5}$;

$x\left(x^{2}+y^{2}\right)+a\left(y^{2}-x^{2}\right) = 0, a \neq 0$;

$(2 a-x) y^{2} = x(x-a)^{2}, a \neq 0$

$\left(x^{2}+y^{2}\right)(y-a)^{2}-b^{2} y^{2} = 0, a > 0, b > 0$;

$y^{2} = x^{3}+a x+b$;

$y^{2} = (x-a)(x-b)(x-c), a \leqslant b \leqslant c$.

$x^{2 / 3}+y^{2 / 3} = 2^{2 / 3}$;

$y^{2} = \sin x^{2}$;

$y^{2} = \sin^{3} x$;

$y^{2} = 1-e^{-x^{2}}$;

$y^{2} = 1-e^{-x^{3}}$;

$y = \begin{cases} x /\left(1+e^{1 / x}\right), & \text{ if } x \neq 0 \\ 0, & \text{ if } x = 0 \end{cases}$

$y = \begin{cases} x \ln x, & \text{ if } x \neq 0 \\ 0, & \text{ if } x = 0 \end{cases}$

$y^{x} = x^{y}$.

$x^{4}+2 x^{2} y-x y^{2}+y^{2} = 0$;

$x^{4}-6 x^{2} y+25 y^{2}-16 x^{2} = 0$;

$\left(x^{2}+y^{2}-6 x\right)^{2}-\left(x^{2}+y^{2}\right) = 0$;

$x^{4}+y^{4}-6 y^{3}+8 x^{2} y = 0$;

$x^{4}+2 y^{3}-4 x^{2} y = 0$;

$x^{5}+y^{5}-x y^{2} = 0$;

$y^{3}-x^{2} y+x^{5} = 0$;

$\left(x^{2}+y^{2}\right)^{3}-27 x^{2} y^{2} = 0$.

$y = C x-C^{2}$;

$y = C x+1 / C$;

$y = C x-\ln C$;

$y = C x+\cos C,\left|C\right| < \pi / 2$

$y = C x+f(C)$, $f$ is a continuously differentiable function;

$x \cos C+y \sin C = p, p > 0$;

$3(C x-y) = C^{3}$;

$2 C^{2}(y-C x) = 1$.

$(x-C)^{2}+y^{2} = R^{2}$;

$(x-C)^{2}+y^{2} = C^{2} / 2$;

$(x-C)^{2}+(y-C)^{2} = C^{2}$

$(x-C)^{2}+y^{2} = R^{2}-C^{2},\left|C\right| \leqslant R / \sqrt{2}$.

$y = \frac{(x+C-1)^{2}}{C}$;

$x = C+\frac{y^{2}}{C}$;

$y = x^{2}-2 C x-3 C^{2}$;

$y^{2} = 2 C x+C^{2} ; C \neq 0$;

$y = C^{2}(x-C)^{2}, C \neq 0$;

$C x^{2}+C^{2} y = 1$.

$y = C^{2} /(x-C)$;

$y = C / x-C^{2}$;

$x = y^{4} /(4 C)+C, C > 0$;

$((y-C) / 3)^{3}+((x+C) / 5)^{5} = 0$;

$y = C e^{x / C}$;

$y = C e^{2 x}+1 / C$;

$y = x+\sin (x+C)$;

$y = C \cdot \operatorname {tg} x-C^{2}$.

$C^{2} y = 4(C+1) x$;

$y = (x-C)^{3}$;

$y^{2} = (x-C)^{3}$;

$y^{3} = (x-C)^{2}$;

$y-C = ((x-C) / 3)^{3}$;

$((y+C) / 5)^{5} = ((x+C) / 7)^{7}$;

$y = 3 C x^{4 / 3}-C^{3}$;

$(2-x)(y-C)^{2} = x^{2}(2+x)$;

$x^{3}+(y-C)^{3} = 3 x(y-C)$;

$\left(y-(x-C)^{2}\right)^{2} = (x-C)^{5}$.

$z = (x-1) \cos C+(y-2) \sin C$;

$x^{2}+y^{2}+(z-C)^{2} = R^{2}$;

$(x-C)^{2}+(y-C)^{2}+(z-C)^{2} = R^{2}$

$(x-C)^{2}+(y-C)^{2}+(z-C)^{2} = C^{2}, C \neq 0$.

on the circle $x^{2}+y^{2} = r^{2}, z = 0$;

on the cone $x^{2}+y^{2} = z^{2}$.