Consider the equation x^2 - 2xy + 4y^2 = 84

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

2) Find the equation of the tangent lines to the curve at the point x=2.

3) Find (d^2 y)/(dx^2) at (0, sqrt(21))

Guest Apr 25, 2019

#1**+2 **

1. Slope at any point ...using implicit differentiation.... 2x - 2y - 2xy ' + 8y y ' = 0

y' ( 8y - 2x ] = 2y - 2x

y ' = [ 2y - 2x ] / [ 8y - 2x ]

2. At x = 2....we can find y as

2^2 - 2(2)y + 4y^2 = 4

4 - 4y + 4y^2 = 84

4y^2 - 4y - 80 = 0

y^2 - y - 20 = 0 factor

(y - 5) (y + 4) = 0 .....so y = 5 or y = -4

So....we have the points ( 2, 5) and (2, -4)

So the slope at (2, 5) = [2(5) - 2(2) ] / [ 8(5) - 2(2) ] = 6/36 = 1/6

So...the equation is y = (1/6) (x - 2) + 5

The slope at (2, -4) = [ 2(-4) - 2(2)] / [ 8(-4) -2(2)] = -12/ -36 = 1/3

So...the equation is y = (1/3)(x - 2) - 4

See the graph here : https://www.desmos.com/calculator/q55m709tum

I don't know the answer to the last one.....sorry !!!

CPhill Apr 26, 2019