Find the length of the perpendicular drawn from the point (4,-7) to the line joining the origin and the point of intersection of the lines 2 x-3 y+14=0 and 5 x+4 y-7=0

Step-by-step solution

Let the two given lines be L₁: 2x - 3y + 14 = 0 and L₂: 5x + 4y - 7 = 0 . To find the point of intersection, multiply L₁ by 4 and L₂ by 3: 8x - 12y + 56 = 0 15x + 12y - 21 = 0 Adding these equations gives 23x + 35 = 0 , so x = - 35 23 . Substituting x into L₁ : 2(- 35 23 ) - 3y + 14 = 0 - 70 23 + 14 = 3y 3y = -70 + 322 23 = 252 23 y = 84 23 . The point of intersection is P(- 35 23 , 84 23 ) . The line joining the origin (0,0) and P has the equation y - 0 = 84/23 - 0 -35/23 - 0 (x - 0) , which simplifies to y = - 84