Suppose x^2+y^2=k^2, this meets the line when:
Explain why the minimum value of x^2+y^2 on x+y=4 occurs at the point at which a graph of x^2+y^2=Constant is tangent to the line x+y=4
which has roots:
x= [8+/-sqrt(64-8(16-k^2))]/4=2 +/- sqrt(-4+k^2/2)
Now for large k this has two solutions but as k is reduced there comes a point
where k^2/2=4, where there is only one root, any smaller and there are no roots.
So the minimum k for which there are roots corresponds to the line being
This seems over complex but at a point of tangency the slope of theThen
Using this answer, and implicit differentiation to find the slope of the circle, find the minimum value of x^2+y^2 such that x+y=4.
circle will be equal to that of the line (-1).
d/dx(x^2+y^2)=2x +2y dy/dx=0,
The slope of the line is -1, so x=y at the point of tangenct, so:
2x=4, so x=2, y=2 and hence the minimum of x^2+y^2=8.