I am helping a friend with a calculus problem he has, he just started his first calculus course while I have done a couple in the past. However, it seems like I'm rusty :(

Here is the problem:

We have an ellipse:

$\displaystyle 9\,{x}^{2}+8\,{\it xy}+25\, \left( y-3 \right) ^{2}=36$

And a point not on the ellipse at (2;2).

We want to know the two points on the ellipse where there is a tangent also going through that point.

First, I have implicitly differentiated the equation of the ellipse to end up with:

$\displaystyle {y' = \frac {18\,x+8\,y}{-8\,x-50\,y+150}}$

Then, since I know that:

$\displaystyle {\frac {2-y}{2-x} = \frac {18\,x+8\,y}{-8\,x-50\,y+150}}$

I found:

$\displaystyle {y = -({\frac {36\,x-18\,{x}^{2}+16\,y-8\,yx}{-8\,x-50\,y+150}}-2)}$

Now I need to solve the following system for x and y:

$\displaystyle 9\,{x}^{2}+8\,{\it xy}+25\, \left( y-3 \right) ^{2}=36$

$\displaystyle {y = -({\frac {36\,x-18\,{x}^{2}+16\,y-8\,yx}{-8\,x-50\,y+150}}-2)}$

But I am unable to. My friend is specifically allowed to use a symbolic calculator to solve the problem, but with Maple I get the following garble:

I have tested the two pairs of solution given with both of the equations in my system and they work. I need help to solve that system here and find the two (x;y) pairs. Thank you a lot in advance.Quote:

1 / / 4 3

{ x = - --------------- \1600 RootOf\1600 _Z - 20000 _Z

\ 18 (32 xy - 65)

2 2

+ (93997 + 512 xy) _Z + (-191958 - 5776 xy) _Z + 576 xy + 141453

\ / 4 3 2

+ 11024 xy/^3 - 14800 RootOf\1600 _Z - 20000 _Z + (93997 + 512 xy) _Z

2 \ /

+ (-191958 - 5776 xy) _Z + 576 xy + 141453 + 11024 xy/^2 + 45897 RootOf\

4 3 2

1600 _Z - 20000 _Z + (93997 + 512 xy) _Z + (-191958 - 5776 xy) _Z

2 \ / 4 3

+ 576 xy + 141453 + 11024 xy/ + 512 RootOf\1600 _Z - 20000 _Z

2 2

+ (93997 + 512 xy) _Z + (-191958 - 5776 xy) _Z + 576 xy + 141453

\ \ / 4 3

+ 11024 xy/ xy - 2888 xy - 45279/, y = RootOf\1600 _Z - 20000 _Z

2 2

+ (93997 + 512 xy) _Z + (-191958 - 5776 xy) _Z + 576 xy + 141453

\\

+ 11024 xy/ }