Given, the point P[{x₁ +  t(x₂ - x₁ )}, {y₁ + t (y₂ - y₁ )}] divides the join of A(x₁ ,y₁ ) and B(x₂ ,y₂ ) internally.

Let the ratio is k : 1

Now, {x₁ +  t(x₂ - x₁ ), y₁ + t (y₂ - y₁ )} = {(kx₂ + x₁ )/(k + 1), (ky₂ + y₁ )/(k + 1)}

Now, x₁ +  t(x₂ - x₁ ) = (kx₂ + x₁ )/(k + 1)

=> x₁ (1 - t) + tx₂ =  kx₂ /(k + 1) + x₁ /(k + 1)

Now, equate coefficient of x₂ , we get

=> t = k/(k + 1)

Since k > 0              {since ratio can not be negative}

=> 0 < k/(k + 1) < 1

=> 0 < t < 1

Hence, the value of t always lies between 0 and 1