use implicit differentiation to find the point where the parabola defined by x^2-2xy+y^2+4x-8y+20=0
Your original question is incomplete.
However, for x^2-2xy+y^2+4x-8y+20=0
2x - 2x(dy/dx) - 2y + 2y(dy/dx) + 4 - 8dy/dx = 0
dy/dx(-2x + 2y - 8) = -2x + 2y - 4
dy/dx = (-2x + 2y - 8)/(-2x + 2y - 4)
= (x - y + 4)/(x - y + 2)
btw, this is what your graph looks like
https://www.wolframalpha.com/input/?i=plot+x%5E2-2xy%2By%5E2%2B4x-8y%2B20%3D0
Now what is your question?
oops - that may have been someone else. Try
https://www.jiskha.com/questions/1805378/Use-implicit-differentiation-to-find-the-points-where-the-parabola-defined-by-x-2-2xy-y-2-8x-4y-20
To find the point on the parabola defined by the equation x^2 - 2xy + y^2 + 4x - 8y + 20 = 0 using implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x.
Keep in mind that when differentiating y with respect to x, you'll need to use the chain rule.
The derivative of x^2 with respect to x is 2x,
The derivative of -2xy with respect to x is -2x(dy/dx) - 2y,
The derivative of y^2 with respect to x is 2y(dy/dx),
The derivative of 4x with respect to x is 4,
The derivative of -8y with respect to x is -8(dy/dx),
And, the derivative of 20 with respect to x is 0 (since 20 is a constant).
Step 2: Simplify the equation.
Combine like terms and move all the terms containing dy/dx to one side of the equation, while keeping the constant terms on the other side.
You should end up with the following equation: (2x - 2y)dy/dx = -4x + 8y - 4
Step 3: Solve for dy/dx.
To find the value of dy/dx, divide both sides of the equation by (2x - 2y).
This will give you the equation: dy/dx = (-4x + 8y - 4)/(2x - 2y)
Step 4: Find the point(s) on the parabola.
To find the points on the parabola, substitute the values for x and y from the original equation into the expression for dy/dx.
This will give you the value of dy/dx at each point on the parabola.
Note: This equation can be simplified further by canceling out the common factors of -4 from the numerator and denominator, resulting in the equation: dy/dx = (-x + 2y + 1)/(x - y)
I already showed you how to find where the parabola has vertical and horizontal tangents. What did you not like about my solution?
Solution for x^2+y^2-4x+8y+20=0 equation:
Simplifying
x2 + y2 + -4x + 8y + 20 = 0
Reorder the terms:
20 + -4x + x2 + 8y + y2 = 0
Solving
20 + -4x + x2 + 8y + y2 = 0
Solving for variable 'x'.
The solution to this equation could not be determined.