 Hi, let us discuss the following question. The question says a person standing at the junction crossing two straight paths represented by the equations 2x minus 3y plus 4 is equal to 0, and 3x plus 4y minus 5 is equal to 0 wants to reach the path whose equation is 6x minus 7y plus 8 is equal to 0 in the least time. Find an equation of path that you should follow. Let's first make a figure to understand this question. Suppose Oa and Ob are the straight paths whose equations are 2x minus 3y plus 4 is equal to 0, and 3x plus 4y minus 5 is equal to 0. These two paths intersect that O, and the person is standing at point O. Now, let Ab be the path where the person wants to reach, and the equation of Ab is 6x minus 7y plus 8 is equal to 0. He has to reach this path in least time. The shortest path from O to the path Ab is perpendicular from O to Ab. So this means we have to find equation of perpendicular path On. Now, keeping all this in mind, let's now begin with the solution. Equation of Oa is 2x minus 3y plus 4 is equal to 0, and equation of Ob is 3x plus 4y minus 5 is equal to 0. By solving these two equations, we can get the coordinates of point O, since O is the point of intersection of path Oa and Ob. So let's now solve these two equations. I'll multiply the first equation by 4 and second by 3. So now we have 8x minus 12y plus 16 is equal to 0, 9x plus 12y minus 15 is equal to 0, and adding these two equations, we get 17x plus 1 is equal to 0. And this implies x is equal to minus 1 by 17. Substituting the value of x in equation of Oa, we get the value of y as 22y17. Now, equation of line Ab is xx minus 7y plus 8 is equal to 0. Slope of this line is 6 by 7. Now, since m is perpendicular to Ab, therefore product of slopes of On and Ab is minus 1, thus slope of Om is minus 7 by 6, right? Now, we will find equation of perpendicular path Om. This path is passing through the point minus 1 by 17 and 22 by 17 and is having slope minus 7 by 6. So equation of Om is y minus 22 by 17 is equal to minus 7 by 6 into x plus 1 by 17. Now, this implies y minus 22 by 7 is equal to minus 7 by 6x minus 7 by 102. Multiplying this equation by 102, we get 102 minus 1 and 32 is equal to minus 119x minus 7. Now, this implies 119x plus 102y is equal to 125. Hence, the required equation of shortest path is 119x plus 102y is equal to 125. This is our required on-tock. So this completes the session. Bye and take care.