Question

The 2x + z = 1 plane when intersecting the paraboloid z = x ^ 2 + y ^ 2 + 1 forms a circle. Find the largest distance of the points on the circle from the coordinate origin

Answer 1

The circle can be obtained by substituting z = 1-2x in the equation of paraboloid

Hence, we can use the parametric form

The function that needs to be maximized is

For purpose of maximization, we can just consider the argument inside the radical sign and maximize that. Also, we shall work in the polar coordinates for simplicity, i.e. find the angle θ

It is not difficult to see that the above function is maximum when θ = 0°, making cos θ = 1. Hence the maximum distance of the point from origin is