Anyone who knows about Descartes' Method??? Find the equation of the tangent line using Descartes' Method!!!!!

Question

please help me with Descartes' Method!!!! thx~~
Find the equation of the tangent line using Descartes' Method!!! plz~ help? Make sure have to use Descartes' Method
i knows the way to do with derivative
Find the equation of the tangent line to the following functions,
using Descartes' Method.

#1 f(x) = x^2, at, (2,4)
#2 f(x) = x^3+7x, at, (1,8)
#3 f(x) = x, at, (-3,3)

Could u guys help me to get tangent line of equation above
3 problems??? have test help thx!!!

about Descartes' Method.
1. (x-xo)^2+y^2 = (x1-xo)^2+(y1)^2
substituting f(x) for y in the above equation, we get:
(x-xo)^2+(f(x))^2 = (x1-xo)^2 + (y1)^2

Answer 1

Try Google or Yahoo. You'll find worked examples there. I think it needs diagrams to understand; they and the accompanying explanations are too complex to attempt to reproduce here.

Answer 2

Using Reference 1 as my guide and using a circle centred on the y-axis rather than than the x-axis .. the idea being that the distance from the centre to the point where the circle cuts the curve equals the distance from the centre to the point. If they are a perfect square then the circle just touches the curve and so the radius is perpendicular to the tangent to the curve

#1 f(x) = x² at (2,4)

(x)² + (y - yo)² = (x1)² + (y1 - yo)² where the circle is centred at (0, yo)

At (2, 4)
(x)² + (y - yo)² = (2)² + (4 - yo)²

Now x² = y

Thus y + y² - 2yoy + yo² - (4 + 16 - 8yo + yo²) = 0

ie y² - (2yo - 1)y + 4(2yo - 5) = 0

For there to be only one value for y (or rather 2 values the same) the discriminant of this quadratic has to equal 0

ie (-(2yo - 1))² - 4 * 1 * 4(2yo - 5) = 0
ie 4yo² - 4yo + 1 + 80 - 32yo = 0
4yo² - 36yo + 81 = 0

ie (2yo - 9)² = 0

yo = 9/2

So the centre of the circle is (0, 9/2)

So the slope of the radius is (4 - 9/2)/(2 - 0)
ie -1/4

So slope of tangent is 4
and its equation is y = 4x - 4

#2

y = x³ + 7x at (1, 8)

Choose circle centred (Xo, 0)

Then (x - Xo)² + (x³ + 7x)² = (1 - Xo)² + 8²

So x² - 2Xox + Xo² + x^6 + 14x^4 + 49x² = 1 - 2Xo + Xo² + 64

x^6 + 14x^4 + 50x² -2Xox + 2Xo - 65 = 0


ie YUCK!!!!!!!! I have no idea where to go from here!!!

Answer 3

Try the implicit function procedure.