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

Hope this information will clear your doubts about the topic.

If you have any more doubts, just ask here on the forum and our experts will try to help you out as soon as
possible.

Regards

  • 1
What are you looking for?