Expert: Paul Klarreich - 3/22/2007

Question
Please  can you solve the following :

1) Find the equation of the tangent to the ellipse

       ( x^2)/9  + y =  1

2) What are the rectangular coordinates of the point (10,
  one hundred and fifty degrees ) ?

3) Which of the following ellipses has its center at the
  point C ( 2, 1 ) ?

a) x^2 +4y^2   -4x  -16y + 4  = 0

0r, ( b), 4x^2  + 9y^2  - 16x -18y  - 11 = 0  

Answer
Questioner:   Alberto
Category:  Calculus
 
Subject:  Calculus
Question:  Please  can you solve the following :

1) Find the equation of the tangent to the ellipse

      ( x^2)/9  + y =  1

2) What are the rectangular coordinates of the point (10,
 one hundred and fifty degrees ) ?

3) Which of the following ellipses has its center at the
 point C ( 2, 1 ) ?

a) x^2 +4y^2   -4x  -16y + 4  = 0

0r, ( b), 4x^2  + 9y^2  - 16x -18y  - 11 = 0
...................................................
Hi, Alberto,

1) Find the equation of the tangent to the ellipse

      ( x^2)/9  + y =  1

I think you are missing something.  The equation you supplied is for a parabola, not an ellipse.  I think you meant:

1) Find the equation of the tangent to the ellipse (x^2)/9  + y^2 = 1, at the point (??,??)

You need that point.  So I will just get you started and you can supply the point later and finish.

Use implicit differentiation:

2x/9 + 2y dy/dx = 0

x/9 + y dy/dx = 0

dy/dx = -x/(9y)

Now use the point (??,??) to get the slope and the equation.

2) What are the rectangular coordinates of the point (10,
 one hundred and fifty degrees ) ?

Use  x = r cos @   and  y = r sin @, with:

r = 10,   @ = 150 degrees.


3) Which of the following ellipses has its center at the
 point C ( 2, 1 ) ?

a) x^2 +4y^2   -4x  -16y + 4  = 0

0r, ( b), 4x^2  + 9y^2  - 16x -18y  - 11 = 0

To find the center of an ellipse,

A. Complete the square to force the equation into the form:
(x-h)^2   (y-k)^2
------- + ------- = 1
 a^2      b^2

B. Then (h,k) are the coordinates of the center.

I'll do this on one of the examples -- the second one.

4x^2  + 9y^2  - 16x - 18y  - 11 = 0

4x^2  - 16x           + 9y^2  - 18y      = 11

4(x^2  - 4x      ) + 9(y^2  - 2y       ) = 11

4(x^2  - 4x  + 4 ) + 9(y^2  - 2y       ) = 11 + 16

4(x^2  - 4x  + 4 ) + 9(y^2  - 2y  + 1  ) = 11 + 16 + 9

4(x - 2)^2         + 9(y - 1)^2  = 36

(x - 2)^2   (y - 1)^2
--------- + ---------- = 1
  9            4

Got the idea, now?