1. Interpretation of the question

A body is projected vertically upward from the surface of the Earth with speed $u=\lambda v_e$ where $v_e$ is the escape velocity from Earth's surface.

We need to find the maximum distance from the center of Earth reached by the body.

2. Use conservation of mechanical energy

Let Earth's radius be $R$ and Earth's mass be $M$.

Initial energy at the surface: $E_i=\frac12 m(\lambda v_e)^2-\frac{GMm}{R}$

At the highest point, velocity becomes zero. If the maximum distance from Earth's center is $r$, then $E_f=-\frac{GMm}{r}$

So, $\frac12 m\lambda^2 v_e^2-\frac{GMm}{R}=-\frac{GMm}{r}$

3. Use escape velocity relation

We know $v_e=\sqrt{\frac{2GM}{R}}$ Rightarrow v_e^2=\frac{2GM}{R}$$

Substitute into the energy equation: $\frac12 m\lambda^2\left(\frac{2GM}{R}\right)-\frac{GMm}{R}=-\frac{GMm}{r}$

This gives $\frac{GMm\lambda^2}{R}-\frac{GMm}{R}=-\frac{GMm}{r}$

$\frac{GMm}{R}(\lambda^2-1)=-\frac{GMm}{r}$

Cancel $GMm$: $\frac{\lambda^2-1}{R}=-\frac{1}{r}$

$\frac{1-\lambda^2}{R}=\frac{1}{r}$

Therefore, $r=\frac{R}{1-\lambda^2}$

4. Match with options

This corresponds to:

$\boxed{\frac{R}{1-\lambda^2}}$

So the correct option is B.

5. Verification of physical condition

For the body to reach a finite maximum distance, we need $\lambda<1$ If $\lambda=1$, then $r\to \infty$, which is escape. This is physically consistent.