It is known that a and B are two fixed points, and the distance ratio from the moving point m to a and B is constant λ

It is known that a and B are two fixed points, and the distance ratio from the moving point m to a and B is constant λ

Let m (x, y) be any point on the track. Let's solve the problem, we get the Ma | MB | MB | MB | MB |; = λ, the coordinates of (x + a) 2 + Y2 (x + a) 2 + Y2 (x | ab | = 2A, then a (- A, 0), a (- A, 0), and B (a, 0). Let m (x, y) be any point on the track. Let's solve the problem, we can get (x + X + a) 2 (1 - λ 2) x2 + (1 - (1 - λ 2) 2 (1 - (1 - λ 2) - 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 (1 - (1 - (1 - λ 2 - (1 - λ 2 - (1 - λ 2) A2 = 0 (2) A2 = 0 (1 straight (2) when λ≠ 1, the trajectory equation of point m is x2 + Y2 + 2A (1 + λ 2) 1 − λ 2x + A2 = 0, the trajectory of point m is a circle with (- A (1 + λ 2) 1 − λ 2, 0) as the center and 2A λ|1 − λ 2 | as the radius

Find the necessary and sufficient condition for the curve of equation y = ax ^ + BX + C to pass through the origin, and how to know whether an equation satisfies the condition of passing through a certain point

A necessary and sufficient condition for the curve of equation y = ax ^ + BX + C to pass through the origin
C = 0 when x = y = 0
Necessary and sufficient condition C = 0
Does an equation satisfy the condition of passing through a point and bring a point into the equation

A necessary and sufficient condition for the curve of equation y = ax * x + BX + C to pass through the origin

Passing through the origin
When x = 0, y = 0
0=a*0*0+b*0+c
c=0
So if and only if C = 0

What is the geometric meaning of curve integral? (type 1, type 2)

The first type is linear density, and the second type is the work of variable force on the curve, so it is also called the curve integral of coordinates, which is also called orthogonal decomposition

Who can explain the geometric meaning of the first and second curve integral and the difference between them in detail?

The first kind of curvilinear integral is the integral of arc length, so the differential element is DS
The second kind of curvilinear integral is the integral of coordinates, so the differential elements are DX, Dy, etc
In general, in the rectangular coordinate plane, DS = the square root of the sum of squares of DX and dy

What is the difference between the first type curve integral and the second type curve integral?
There are also surface integrals of type one and type two~

The first kind of curve is the length of the curve, the second kind is the X, y coordinates. How to understand it? Tell you the linear density of a line, ask you the quality of the line, use the first kind. Tell you the path curve equation, tell you the force in X, y directions, work, use the second kind. The second kind of curve can also separate x, y, so it is not difficult to understand the product of the first and second kind of curve

Find the second derivative Y "of x = a (t-sint) y = a (1-cost). The answer is - 1 / a (1-cost) (1-cost)
I need specific problem-solving process, the more detailed the better! Thank you very much

This is the problem of parameter derivation
Because y '= dy / DX = (dy / DT) / (DX / DT)
y''=dy'/dx=(dy'/dt)/(dx/dt)
dx/dt=a(1-cost)
dy/dt=asint
y'=sint/(1-cost)
y''=-1/[a*(1-cost)^2]

The second derivative d ^ 2Y / DX ^ 2 of x = lnsint y = cost + TSINT

Note: dy / DX = y '(T) / X' (T) suppose: z = dy / dxz = dy / DX = y '(T) / X' (T) = t cos (T) / cot (T) = t sin (T) DZ / DX = Z '(t) / X' (T) = (t cos (T) + sin (T)) / cot (T) = sin (T) (T + Tan (T)), so d ^ 2Y / DX ^ 2 = sin (T) (T + Tan (T)

According to the second derivative, the concavity and convexity of cycloid x = a (t-sint), y = a (1-cost) (a > 0) are studied

In fact, it is to find the second derivative of the above parametric equation

Let the parametric equation x = t (1-sint); y = tcost, a be a constant, and find the second derivative d ^ 2Y / (DX ^ 2)

dy/dx=(dy/dt)/(dx/dt)=(cost-tsint)/(1-sint-tcost)d^2y/dx^2=(dy/dx)/(dx/dt)=[(-sint-sint-tcost)(1-sint-tcost)-(cost-tsint)(-cost-cost+tsint)/(1-sint-tcost)^3=[(2sint+tcost)(sint+tcost-1)+(cost-tsint)(2...