At what point do the curves r₁ (t) = t, 2 - t, 24 + t² and r₂ (s) = 6 - s, s - 4, s² intersect?

Solution:

Step 1: Set the pair of components equal to each other that is r\(_1\) (t) = r\(_2\) (s) (since they intersect at a point)

t = 6 - s or t + s = 6 ------> 1

2 - t = s - 4 ------> 2

24 + t² = s² or s² - t² = 24 ------> 3

Step 2: Simplify equation 3 using the value of t from equation 1.

24 + ( 6 - s )² = s²

24 + 36 - 12s + s² = s²

12s = 60

s = 5

t = 1

Step 3: Substitute the values of ‘t’ and ‘s’ in equations (1), (2) and (3).

r\(_1\) (1) = r\(_2\) (5) = (1, 1, 25)

At what point do the curves r\(_1\) (t) = t, 2 - t, 24 + t^² and r\(_2\) (s) = 6 - s, s - 4, s^² intersect?

Summary:

The point where the curves r\(_1\) (t) = t, 2 - t, 24 + t² and r\(_2\) (s) = 6 - s, s - 4, s² intersect is (1, 1, 25).