InterviewSolution
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InterviewSolution
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Consider the linear equations ax+by+cz=0, bx+cy+az=0 and cx+ay+bz=0. Match the conditions/expressions in Column I with statements in Column II. |
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Answer» `Delta = |{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}|` `""= -(a+b+c)(a^(2)+b^(2)+c^(2)-ab-bc-ca)` `""= -(1)/(2)(a+b+c)[(a-b)^(2)+ (b-c)^(2)+ (c-a)^(2)]` a. `a+b+c ne 0` and `""a^(2)+b^(2)+c^(2)-ab-bc-ca=0` or `""(a-b)^(2)+ (b-c)^(2)+ (c-a)^(2)=0` or `""a=b=c` THEREFORE, this question represents identical planes. b. `a+b+c=0` and `""a^(2)+b^(2)+c^(2)-ab-bc-ca ne 0` This means `Delta = 0 and a, b and c` are not all equal. Therefore, all equations are not identical but have infinite solutions. Hence, `""ax+by= (a+b)Z ""` (using `a+b+c=0`) and `""bx+cy= (b+c)z` `rArr""(b^(2)-ac)y=(b^(2)-ac)z rArr y=z` rArr `""ax+by +cy=0 rArr ax=ay` `rArr""x=y=z` Therefore, the equations represent the line `x=y=z`. c. `a+b+c ne 0 and a^(2)+b^(2)+c^(2)-ab-bc-ca ne 0` `rArr"" Delta ne 0` and the equations have only trivial solution, i.e., `x=y=z=0`. Therefore, the equations represent the planes meeting at a single point, namely origin. d. `a+b+c=0 and a^(2)+b^(2)+c^(2)-ab-bc-ca=0` `rArr"" a=b=c and Delta =0 rArr a=b =c =0` `rArr""` All equations are satisfied by all `x, y and z`. `rArr ""` The equations represent the whole of the three-dimensional space. |
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