We know that PA ⊥ Pb, Pb ⊥ PC, PC ⊥ PA in p-abc, and PA = Pb = PC = a, then we find the volume of this pyramid I want the specific process. I must

We know that PA ⊥ Pb, Pb ⊥ PC, PC ⊥ PA in p-abc, and PA = Pb = PC = a, then we find the volume of this pyramid I want the specific process. I must

PA⊥PB,PB⊥PC,PC⊥PA
The three sides are perpendicular to each other
Think of one side as the bottom and the other as the high
Then the volume of triangular pyramid is 1 / 3 * 1 / 2 * a ^ 2 * a = 1 / 6 * a ^ 3

Height: the tangent equation of curve X ^ 2 + y ^ 2 = 5, z = x ^ 2-y ^ 2 at point (1,2, - 3) is
The tangent equation of curve X ^ 2 + y ^ 2 = 5, z = x ^ 2-y ^ 2 at point (1,2, - 3) is

x^2+y^2=5
Normal direction at any point: (x, y, 0)
z=x^2-y^2 => x^2 - y^2 - z=0
Normal direction at any point: (2x, 2Y, - 1)
By substituting (1,2, - 3), two normal directions are obtained
{1,2,0} and {2, - 4, - 1}
The direction vector of straight line is obtained by cross multiplication: {- 2,1, - 8}
Straight line passing point: (1,2, - 3)
The equation obtained by the point formula is as follows:
(x-1)/-2 = y-2 = (z+3)/-8

A problem of tangent equation of curve of higher number
Let f (x) = - x (x-m) be the square of the function. When m = 1, the equation for finding the tangent of the curve y = f (x) at point (2, f (2)) is obtained

m=1
f(x)=-x(x-1)^2=-x(x^2-2x+1)=-x^3+2x^2-x
f'(x)=-3x^2+4x-1
f'(2)=-3*4+8-1=-5
f(2)=-8+8-2=-2
That is, the tangent point is (2, - 2) and the slope is - 5
The equation is y + 2 = - 5 (X-2)
That is: y = - 5x + 8

Urgent: as long as the result, the tangent equation of the curve y = x + arctanx at x = 0 is?

The derivative of y = 1 + 1 / (1 + x ^ 2) when x = 0, the derivative of y = 2, y = 0
So the tangent is y-0 = 2 (x-0), that is y = 2x