if the points (a,b), (a1,b1) and (a-a1,b-b1) are collinear show that ab1=a1b. also show that the line joining the given points passes through the origin 

Dear student,

Please find below the solution to the asked query:

Consider the following points.  Aa,b,  Ba1,b1,  Ca-a1,b-b1Since the given points are collinear, we have areaABC=0First find the area of areaABC as follows:  areaABC=12x1y2-y3+x2y3-y1+x3y1-y2                        =12ab1-b-b1+a1b-b1-b+a-a1b-b1                        =12ab1-b+b1+a1b-b1-b+ab-b1-a1b-b1                        =12-ab-a1b1+ab-ab1+a1b+a1b1                        =12-ab1-a1b                        =12ab1-a1bThis gives,   ab1-a1b=0     ab1=a1bNow to show that this line passes through origin 0,0, we need to show thatthe points  O0,0, Aa,b,  Ba1,b1 are collinear     AreaOAB=120b-b1+ab1-0+a10-b                             =120+ab1-a1b                             =12ab1-a1b                             =12×0                             =0This means that O0,0, Aa,b,  Ba1,b1 are collinearSo the given line passes through origin O0,0

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