Tangent plane equation and normal equation of surface z = x ^ 2 + 5xy-2y ^ 2 at point (1,2,3)

Tangent plane equation and normal equation of surface z = x ^ 2 + 5xy-2y ^ 2 at point (1,2,3)

Take partial derivatives of X, y, Z, f'x = 2x + 5Y, f'y = 5x-4y, f'z = - 1, substitute x, y, Z, f'x = 12, f'y = - 3, f'z = - 1, normal equation is (x-1) / 12 = (Y-2) / - 3 = (Z-3) / - 1, tangent plane equation is 12 (x-1) - 3 (Y-2) - (Z-3) = 0

The following bivariate quadratic equations are decomposed into two bivariate quadratic equations (1) x-3xy + 2Y, which are the square of x-3xy + 2Y____ And______
(2) The square of X + the square of 2xy-3y = 0_______ And_______

The following quadratic equations of two variables are decomposed into two quadratic equations of two variables
(1) The square of x-3xy + 2Y = 0 is decomposed into x-2y = 0____ And__ x-y=0____
(2) The square of X + the square of 2xy-3y = 0_ x+3y=0 ______ And___ x-y=0____

The binary quadratic equation x * 2-y * 2-2x + 2Y = 0 is transformed into two linear equations
So what are the two first-order equations

x²-y²-2x+2y=0
∴（x²-2x+1)-(y²-2y+1)=0
∴(x-1)²-(y-1)²=0
∴(x-1+y-1)(x-1-y+1)=0
∴(x+y-2)(x-y)=0
Ψ X-Y = 0 or x + Y-2 = 0
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What kinds of cross sections can be obtained by cutting a cube with a plane?

Triangle (a line passing through the area within the diagonal of a vertex and the opposite face)
Rectangle (through two opposite edges or one edge)
Square (parallel to a face)
Pentagon (passing a point on four edges and a vertex or a point on five edges)
Hexagon (passing through points on six edges)
Regular hexagon (passing through the midpoint of six edges)
Diamond (passing relative vertex)
Trapezoid (a line of unequal length passing through two opposite planes)