If the orthocenter of the triangle formed by the lines y = x + 1, y = 4x − 8 and y = mx + c is at (3, −1), then m − c is:
❓ Question If the orthocenter of the triangle formed by the lines y = x + 1 , y = 4 x − 8 , y = m x + c is at the point ( 3 , − 1 ) (3,-1) , then find the value of m − c m - c . 🖼️ Question Image ✍️ Short Solution Step 1 — Find intersection of the first two lines (one vertex). Solve x + 1 = 4 x − 8 x+1 = 4x - 8 . Move terms: 1 + 8 = 4 x − x 1 + 8 = 4x - x → 9 = 3 x 9 = 3x → x = 3 x = 3 . Then y = x + 1 = 3 + 1 = 4 y = x + 1 = 3 + 1 = 4 . So vertex A = ( 3 , 4 ) A = (3,4) . Step 2 — Use the altitude through A A . An altitude from vertex A A passes through the orthocenter ( 3 , − 1 ) (3,-1) . The line through A ( 3 , 4 ) A(3,4) and orthocenter ( 3 , − 1 ) (3,-1) has the same x x -coordinate, so it is the vertical line x = 3 x = 3 . That altitude is vertical, so the side it is perpendicular to must be horizontal. Therefore the third side y = m x + c y = mx + c must be horizontal — i.e. its slope m = 0 m = 0 . So the third line is y = c y ...