how do you maximize the area of an isosceles triangle that is inscribed within a circle using lagrange multipliers method?

Question

An isosceles triangle is inscribed in a circle of radius r. Find the maximum possible area of the triangle.

We are supposed to use Lagrange Multipliers for this problem. Here is what i tried:

I fixed a point on the circle at (0,r) and then chose another point (x,y) on the circle to form the isosceles triangle with vertices (0,r), (x,y), and (-x,y). Then the area of this triangle is 1/2 x base x height. The base = 2x. The height = r-y, so A(x,y) = x (r-y) = area of triangle formed by the 3 above vertices. In order to use Lagrange Multipliers method, I also need to figure out my constraint function. In this case, it is the circle $x^2+y^2=r^2$. Therefore my constraint function $g(x)=x^2+y^2$

Next step is to compute partial derivatives:

$A_x=r-y$

$A_y=-x$

$g_x=2x$

$g_y=2y$

This will give me my equations to solve:

$r-y = 2\lambda x$

$-x = 2\lambda y$

$x^2+y^2=r^2$

Here is where I am stuck. How do you solve this system?

Answer 1

[This should have been a comment rather than an answer, but the writer appears to need a higher MSE reputation to post a comment.]

You currently have a scheme with 3 functions and 4 variables, it cannot be solved. Is the radius known by any chance? This would make the problem solvable.