Find the points on the cone z2 = x2 + y2 that are closest to the point (6. 2. 0).

Answer

assume the required point is Q(x, y, z), or Q(x, y,±(x²
+ y²))
the square of the distance between Q and P(6, 2, 0) is
F(x, y) = (x - 6)² + (y - 2)² +
x²+ y²)
= 2x² + 2y² - 12x - 4y + 40
= 2(x² + y² - 6x - 2y + 20)
F_x = 2(2x - 6) = 0, x = 3
F_y = 2(2y - 2) = 0, y = 1
so Q is (3, 1, ±√10)