Find the tangent plane and normal equation of the surface X & # 178; + 2Y & # 178; + 3Z & # 178; = 21 at the point (- 1, - 2,2)

Find the tangent plane and normal equation of the surface X & # 178; + 2Y & # 178; + 3Z & # 178; = 21 at the point (- 1, - 2,2)

The answer is this,
one
X ^ 2 + 2Y ^ 2 + 3Z = 21 normal vector at a point (2x, 4Y, 3)
So the discovery vector at (1, - 2,2) = (2, - 8,3)
So we find the equation: (x-1) / 2 = (y + 2) / - 8 = (Z-2) / 3
two
Direct nested formula
D = | 1-1 + 1 + 2 | / radical 3 = radical 3

How to set the equation of known space surface in the tangent plane equation of any point?

Let f (x, y, z) = 0
Then the normal vector of its tangent plane at points (x0, Y0, Z0) can be expressed as
n0=(F'x(x0,y0,z0),F'y(x0,y0,z0),F'z(x0,y0,z0))
So the tangent plane equation is
F'x(x0,y0,z0) (x-x0)+F'y(x0,y0,z0) (y-y0)+F'z(x0,y0,z0) (z-z0)=0

Is there a positive integer solution to the equation 2x ^ 2-3xy-xy ^ 2 = 98? Yes, ask for a solution; if not, explain the reason
By the way
It is known that a, B and C are the three sides of the triangle ABC. Prove: (a ^ 2 + B ^ 2-C ^ 2) ^ 2-4a ^ 2B ^ 2

Because 2x ^ 2-3xy-xy ^ 2 = 98, the factor of X (2x-3y-y & sup2;) = 98 is only 1,2,7,7,98, that is, when x may be equal to 1,2,7,98, x = 1, y has no solution, x = 2, y has no solution, x = 7, y = 0 or - 3, x = 98, y has no positive integer solution

Finding the integer solution of the Diophantine equation x ^ 2 + 3xy + 2Y ^ 2 + 2x-y = 25

(x, y) are as follows:
-8,1
3,-5
3,1
10,-5
12,-11
The image is hyperbolic and the solution is hyperbolic
y = (1/4)[-3x + 1 ± √(x² - 22x + 201)]