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# 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?

**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 + t^{2} = s^{2} or s^{2} - t^{2} = 24 ------> 3

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

24 + ( 6 - s )^{2} = s^{2}

24 + 36 - 12s + s^{2} = s^{2}

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 + t^{2} and r\(_2\) (s) = 6 - s, s - 4, s^{2} intersect is (1, 1, 25).

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