Find the point (x, y) of x2+14xy+49y2=100 that is closest to the origin and lies in the first quadrant.

Notice that

so the constraint is a set of two lines,

and only the first line passes through the first quadrant.

The distance between any point in the plane is , but we know that and share the same critical points, so we need only worry about minimizing . The Lagrangian for this problem is then

with partial derivatives (set equal to 0)

We have

which tells us that

so that is a critical point. The Hessian for the target function is

which is positive definite for all , so the critical point is the site of a minimum. The minimum distance itself (which we don't seem to care about for this problem, but we might as well state it) is .

it should be a and d. lmk if i'm wrong xd