Sliding Lines

Calculus Level 4

Two lines have the following equations (assuming \( p = (x, y) \)):

\( n_1 \cdot (p - p_1(t) ) = 0 \)

with \(n_1 = (1, 2) , p_1(t) = (1, 1) + (1, 3) t \) , and

\( n_2 \cdot (p - p_2(t) ) = 0 \)

with \( n_2 = (1, -1), p_2(t) = (-1,1) + (1, -1) t \)

\( t \) is the time parameter. The intersection of these two lines traces a line (shown in red in the animation). If the slope of the traced line is \( \dfrac{a}{b} \), where \( a \) and \( b \) are coprime positive integers, find \( a + b \)

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