# At what point do the curves r_{1} (t) = t, 4 - t, 35 + t^{2} and r_{2} (s) = 7 - s, s - 3, s^{2} 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 + t^{2} = s^{2} or s^{2} - t^{2} = 35 ------> 3

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

35 + ( 7 - s )^{2} = s^{2}

35 + 49 - 14s + s^{2} = s^{2}

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 r_{2} (s) = 7 - s, s - 3, s^{²} intersect?

**Summary:**

The point where the curves r\(_1\) (t) = t, 4 - t, 35 + t^{2} and r\(_2\) (s) = 7 - s, s - 3, s^{2} intersect is (1, 3, 36)