At what point do the curves r₁ (t) = t, 4 - t, 35 + t² and r₂ (s) = 7 - s, s - 3, s² intersect?

Solution:

Step 1: Set the pair of components equal to each other that isr\(_1\) (t) =r\(_2\) (s)

t = 7 - s or t + s = 7 ------> 1

4 - t = s - 3 ------> 2

35 + t² = s² or s² - t² = 35 ------> 3

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

35 + ( 7 - s )² = s²

35 + 49 - 14s + s² = s²

14s = 84

s = 6

t = 1

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

r\(_1\) (1) = r\(_2\) (6) = (1, 3, 36)

At what point do the curves r1 (t) = t, 4 - t, 35 + t^² and r₂ (s) = 7 - s, s - 3, s^² intersect?

Summary:

The point where the curves r\(_1\) (t) = t, 4 - t, 35 + t² and r\(_2\) (s) = 7 - s, s - 3, s² intersect is (1, 3, 36)