Problem Statement:

The distance between the origin and the point nearest to it on the surface z² = 1 - xy is

a) 1
b) 3/2
c) 3
d) -2

Solution:

Given, the surface equation is z² = 1 - xy

Let us consider the distance between the origin and a point (x, y, z) on the surface.

The distance formula between the two points is given by:

d = √((x-0)² + (y-0)² + (z-0)²) = √(x² + y² + z²) ---(1)

Now, we need to find the point on the surface which is nearest to the origin. Let (x₀, y₀, z₀) be that point.

Then, the distance from the origin to that point is minimum. Thus, the distance from the origin to any other point (x, y, z) on the surface is greater than or equal to d.

We can write the above statement in mathematical notation as follows:

√(x² + y² + z²) ≥ √(x₀² + y₀² + z₀²) ---(2)

Now, let us substitute the given surface equation into equation (2) and simplify it.

√(x² + y² + (1-xy)) ≥ √(x₀² + y₀² + (1-x₀y₀)) ---(3)

Squaring both sides of equation (3), we get:

x² + y² + (1-xy) ≥ x₀² + y₀² + (1-x₀y₀)

Rearranging the terms, we get:

x₀² + y₀² - xy₀ ≥ 1 - x - y + xy

Now, the left-hand side of the above inequality is a quadratic expression in x or y.

Let us take partial derivatives of the left-hand side with respect to x and y and equate them to zero to find the minimum value of the left-hand side.

∂/∂x (x₀² + y₀² - xy₀) = -y₀ = 0
∂/∂y (x₀² + y₀² - xy₀) = -x₀ = 0

Solving the above two equations, we get:

x₀ = y₀ = 0

Thus, the point nearest to the origin on the surface is (0, 0, 1). Substituting this point into equation (1), we get:

d = √(0² + 0² + 1²) = 1

Therefore, the distance between the origin and the point nearest to it on the surface z² = 1 - xy is 1.

Hence, the correct option is (a) 1.