Solve a system of equations

How can I find the x and y of the point with minimum z which lies on the intersection of a sphere and a plane? Sphere's equation: x^2+y^2+z^2=16 Plane's equation: x+y+z=0

[Ryan Takatsuka]

To find the intersection of these 2 equations, they must be equated to each other. Solve the plane equation for z and square the equation:

z = -x - y

z^2 = x^2 + 2xy + y^2

Now solve the sphere equation for z^2:

z^2 = 16 - y^2 - x^2

Now these equations can be equated to each other:

x^2 + 2xy + y^2 = 16 - y^2 - x^2

Solving for y results in the equation of an ellipse (it makes sense that the intersection of a sphere and plane would be an ellipse):

y = 0.5 * (-sqrt(32-3x^2) - x)

y = 0.5 (sqrt(32-3x^2) - x)

Now just plug this equation back into the original plane or sphere equation to find all of the z values. Then find the minimum.

% Create The x vector
x = linspace(-5,5,10000);

% The ellipse equation (positive and negative versions)
y_pos = 0.5 * (sqrt(32 - 3*x.^2) - x);
y_neg = 0.5 * (-sqrt(32 - 3*x.^2) - x);

% Remove the complex values (no imaginary values)
k = 1;
for i=1:length(x)
  if isreal(y_pos(i))
    y_real(k) = y_pos(i);
    x_real(k) = x(i);
    k = k+1;
  end
end
for i=1:length(x)
  if isreal(y_neg(i))
    y_real(k) = y_neg(i);
    x_real(k) = x(i);
    k = k+1;
  end
end

% Plot the ellipse
figure
scatter(x_real, y_real)

% Calculate the z values
z = -x_real - y_real;

% Find the minimum
min(z)