SOLUTION: Find the equations of the tangent line and normal line to the graph of 6x^2-2xy+y^3=9 at the point (2,-3)

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Question 554379: Find the equations of the tangent line and normal line to the graph of 6x^2-2xy+y^3=9 at the point (2,-3)
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
6x² - 2xy + y³ = 9

Find the derivative implicitly term by term:

12x - 2(x%28dy%29%2F%28dx%29 + y) + 3y²%28dy%29%2F%28dx%29 = 0

Solve for %28dy%29%2F%28dx%29

12x - 2x%28dy%29%2F%28dx%29 - 2y + 3y²%28dy%29%2F%28dx%29 = 0

Isolate the terms in %28dy%29%2F%28dx%29 

-2x%28dy%29%2F%28dx%29 + 3y²%28dy%29%2F%28dx%29 = -12x + 2y

%28dy%29%2F%28dx%29(-2x + 3y²) = -12x + 2y

%28dy%29%2F%28dx%29 = %28-12x%2B2y%29%2F%28-2x%2B3y%5E2%29

Write the positive terms first

%28dy%29%2F%28dx%29 = %282y-12x%29%2F%283y%5E2-2x%29

 = %282%28-3%29-12%282%29%29%2F%283%28-3%29%5E2-2%282%29%29 = %28-6-24%29%2F%283%289%29-4%29 = %28-30%29%2F%2827-4%29 = %28-30%29%2F23 = -30%2F23
    
Finding the tangent line is the algebra problem of finding the equation 
of the line through the point (2,-3) with a slope of -30%2F23

The equation of the tangent line at (2,-3) is

y = -30%2F23x - 9%2F23

Finding the normal line is the algebra problem of finding the equation 
of the line through the point (2,-3) with a slope of 23%2F30

The equation of the normal line at (2,-3) is

y = 23%2F30x - 68%2F15

Edwin