Expert: Paul Klarreich - 12/14/2008

Question
let y = f(x) be the continuous function that satisfies the
equation x^4 - 5(x^2)(y^2) + 4y^4 = 0 and whose graph
contains the points (2,1) and (-2,-2). Let L be the line
tangent to the graph of f at x = 2

a. find an expression for y'
b. Write an equation for line L
c. Give coordinates of a point that is on the graph of f
but is not on line L
d. Give the coordinates of a point that is on line L but is
not on the graph of f

Answer
Questioner:   Chris
Category:  Calculus
Private:  No
 
Subject:  Calculus curve and derivative
Question:  let y = f(x) be the continuous function that satisfies the equation x^4 - 5(x^2)(y^2) + 4y^4 = 0 and whose graph
contains the points (2,1) and (-2,-2). Let L be the line
tangent to the graph of f at x = 2

a. find an expression for y'
b. Write an equation for line L
c. Give coordinates of a point that is on the graph of f
but is not on line L
d. Give the coordinates of a point that is on line L but is
not on the graph of f
..............................................
Hi, Chris,

If x^4 - 5 x^2 y^2 + 4y^4 = 0,
just use Implicit Differentiation.

You will get dy/dx in terms of x and y. (that's how it works.)

Use the point (2,1) [see note below]

You will get a value for dy/dx, and this is your 'm'.

Now use this m, x0 = 2,  y0 = 1, in the point-slope form:

y - y0 = m(x - x0).

and that is your L.


c. Give coordinates of a point that is on the graph of f
but is not on line L.

>> Should not be too hard, once you have the equation for L.

d. Give the coordinates of a point that is on line L but is
not on the graph of f

>> likewise.

Note:  Your equation only has even powers of x and y, so it has all kinds of symmetry.  In fact, the problem should say:

whose graph contains the points (+-2,+-1) and (+-2,+-2)


If you get stuck, send me what you did and I'll try to help.