Jul 1, 2026
Your calculation is correct. Let me show you a simplifying technique.
For example D = O + 100(X - O) + r(Y - O) = (1 - 100 - r)O + 100 X + r Y = d O + 100 X + r Y, where d = 1 - 100 = r.
D is on the line determined by A = a O + 90 X and B = b O + 110 X + 10 Y if and only if A B D = 0. We have \begin{aligned} AB &= (a O + 90 X)(b O + 110 X + 10 Y) \\ AB &= a O(b O + 110 X + 10 Y) + 90 X(b O + 110 X + 10 Y) \\ &= 110a O X + 10a O Y + 90b X O + 900 X Y \\ \end{aligned} and \begin{aligned} A B D &= (110 \_O X + 10 \_O Y + 90 X\_O + 900 X Y)D \\ &= (110 \_O X + 10 \_O Y + 90 X\_O + 900 X Y)(\_O + 100 X + r Y) \\ &= 110 r \_O X Y + 1000 \_O Y X + 900 X Y \_O \\ &= (100 r - 1000 + 900)\_O X Y \\ \end{aligned}