question_answer

Show that the points                                                                             \[(a+5,\,\,a-4),\]                                                                                \[(a-2,\,\,a+3)\]  and (a, a) do not lie on a straight line for any value of a.

Answer:

Given points are \[(a+5,\,\,a-4),\] \[(a-2,\,\,a+3)\]  and (a, a). Now, consider             \[[applying\,\,{{R}_{2}}\to {{R}_{2}}-{{R}_{1}}\,\,and\,\,{{R}_{3}}\to {{R}_{3}}-{{R}_{1}}]\] \[=\frac{1}{2}|[-\,28+35]|\]      [expanding along \[{{C}_{3}}\] ] \[=\frac{7}{2}\ne 0\] Which is also independent of a. Hence, the given points form a triangle, i.e. given points do not lie on a straight line for any value of a          Hence proved.