Final answer:
The point on the hyperbola xy−8=0 closest to (3,0) is obtained by solving the Lagrangian system of equations: 2(x−3)+λy=0, 2y+λx=0, and xy−8=0.
Step-by-step explanation:
To find the point on the hyperbola xy−8=0 closest to the point (3,0), we can use the method of Lagrange multipliers.
The equation of the hyperbola is xy−8=0, which can also be written as xy=8.
Let f(x,y)=(x−3)2+y2 be the square of the distance between the points (x,y) on the hyperbola and (3,0). We want to minimize this distance.
Now, we form the Lagrangian function:
L(x,y,λ)=(x−3)2 +y2+λ(xy−8)
Next, we take partial derivatives with respect to x, y, and λ and set them equal to zero to find critical points:
∂L/∂x=2(x−3)+λy=0
∂L/∂y=2y+λx=0
∂L/∂λ=xy−8=0
Solving this system of equations will give us the coordinates (x,y) of the point on the hyperbola closest to (3,0).
It's a bit complex to solve these equations algebraically, and numerical methods may be needed.
Alternatively, you can use computational tools like Python with libraries such as SymPy or SciPy to solve this system of equations and find the coordinates of the closest point on the hyperbola.