P3 Differentiation/Circles!

Hi...
i dunno how to solve these questions:

1) given that e^2x + e^2y=xy, find dy/dx.

2) The circle C has equation x2 + y2 – 8x – 16y – 209 = 0.

(a) Find the coordinates of the centre of C and the radius of C.
Centre is (4,8), radius 17.

The point P(x, y) lies on C.

(b) Find, in terms of x and y, the gradient of the tangent to C at P.
dy/dx=(4-x)/(y-8)

(c) Hence or otherwise, find an equation of the tangent to C at the point (21, 8).

part c...i dont understand why the tangent is vertical... can somebody tell me why?!!

plus chapter 3 in the heinemann book, what do i need to know exactly?! (edexcel)

thanks guys!!!!!!!

Reply 1

idiopathic

1) given that e^2x + e^2y=xy, find dy/dx.

differentiate each bit w.r.t x

differentiate e^(2x) using the chain rule
differentiate e^(2y) using the chain rule and implicitly
differentiate xy using the product rule and implicitly

so you end up with:
2e^(2x) + 2(dy/dx)e^2y = 1(y) + 1(dy/dx)x
then rearrange for dy/dx

edit: previous thread on that question http://www.thestudentroom.co.uk/showthread.php?p=1813856#post1813856

Reply 2

yazan_l

2) The circle C has equation x2 + y2 – 8x – 16y – 209 = 0.

(a) Find the coordinates of the centre of C and the radius of C.
Centre is (4,8), radius 17.

x² + y² – 8x – 16y – 209 = 0
x² – 8x + y² – 16y = 209
x² – 8x + 16 + y² – 16y + 64 = 209 + 16 + 64 ( you know how to complete the square , right? )
(x - 4)² + (y-8)² = 289
so centre: (4,8)
radius = sqrt[289] = 17

The point P(x, y) lies on C.

(b) Find, in terms of x and y, the gradient of the tangent to C at P.
dy/dx=(4-x)/(y-8)

x² + y² – 8x – 16y – 209 = 0
2x + 2y dy/dx - 8 - 16 dy/dx = 0
(2y - 16) dy/dx = 8 - 2x = 2(4 - x)
dy/dx = 2(4-x) / 2(y-8) = (4-x)/(y-8)

y - y1 = m ( x - x1)
dy/dx at (21, 8) = (4-21)/(8-8) = -17/0 = infinity ==> parallel to y-axis ==> vertical

==> tangent : y= 8

Reply 3

ossoss87

thanks alot... but no one answerd my third question... wt do i neeed to know exactly in chapter 3?!!?

Reply 4

yazan_l

ossoss87

thanks alot... but no one answerd my third question... wt do i neeed to know exactly in chapter 3?!!?

well i dont know what is ch3 to tel u what do u need to know in it!

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