Tangent Curves(important... i need help by Oct. 15)

[Gwennypoo]

consider the cruve defined by x^2 +xy + y^2 = 27

A. Write an expression for the slope of the curve at any point (x,y)

B. Determine whether the lines tangent to the curve at the x-intercepts of the curve are parallel. Show the analysis that leads to your conclusion.

C. Find the points on the curve where lines tangent to curves are vertical.


i do not know whether or not to use implicit differentation or just manipulate it to solve for y.

when using implicit, i get the equation to be -4x-2... and then manipulating gives me square root of (27) - x..

please help with any part!!

 

[galactus]

You can solve this for y and differentiate.

But implicitly:

\(\displaystyle \L\\x^{2}+xy+y^{2}=27\)

\(\displaystyle \L\\2x+x\frac{dy}{dx}+y+2y\frac{dy}{dx}=0\)

\(\displaystyle \L\\\frac{dy}{dx}=\frac{-(2x+y)}{x+2y}\)

For the vertical tangents. What makes the derivative = 0?. x = -2y

Sub that into the original and solve for y.

\(\displaystyle \L\\(-2y)^{2}+(-2y)y+y^{2}=27\)

That'll be the y-coordinates of your points of vertical tangency.

You can easily find x then.