If The Lines Ax+by+c0bx+cy+a0 And Cx+ay+b0 Be Concurrent Class 11 ...

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AnswerQuestion Answers for Class 12Class 12 BiologyClass 12 ChemistryClass 12 EnglishClass 12 MathsClass 12 PhysicsClass 12 Social ScienceClass 12 Business StudiesClass 12 EconomicsQuestion Answers for Class 11Class 11 EconomicsClass 11 Computer ScienceClass 11 BiologyClass 11 ChemistryClass 11 EnglishClass 11 MathsClass 11 PhysicsClass 11 Social ScienceClass 11 AccountancyClass 11 Business StudiesQuestion Answers for Class 10Class 10 ScienceClass 10 EnglishClass 10 MathsClass 10 Social ScienceClass 10 General KnowledgeQuestion Answers for Class 9Class 9 General KnowledgeClass 9 ScienceClass 9 EnglishClass 9 MathsClass 9 Social ScienceQuestion Answers for Class 8Class 8 ScienceClass 8 EnglishClass 8 MathsClass 8 Social ScienceQuestion Answers for Class 7Class 7 ScienceClass 7 EnglishClass 7 MathsClass 7 Social ScienceQuestion Answers for Class 6Class 6 ScienceClass 6 EnglishClass 6 MathsClass 6 Social ScienceQuestion Answers for Class 5Class 5 ScienceClass 5 EnglishClass 5 MathsClass 5 Social ScienceQuestion Answers for Class 4Class 4 ScienceClass 4 EnglishClass 4 MathsIf the lines $ax+by+c=0,bx+cy+a=0$ and $cx+ay+b=0$ be concurrent, then$A){{a}^{3}}+{{b}^{3}}+{{c}^{3}}+3abc=0$ $B){{a}^{3}}+{{b}^{3}}+{{c}^{3}}-abc=0$$C){{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=0$$D)$None of the aboveAnswerVerified506.4k+ viewsHint: To solve the question we need to have the knowledge of types of lines. The solution will start with knowledge of concurrent lines. Lines are said to be concurrent if more than two lines pass through a common point. To check whether the lines are concurrent or not the determinant should be zero. In this question we will equate the determinant with zero. Complete step by step answer:The question asks us to find the relation between a, b, c if the three lines $ax+by+c=0,bx+cy+a=0$ and $cx+ay+b=0$ which are given to us are concurrent or not. The steps that should be used to solve this question is that, as we know that for lines to be concurrent the determinant of the concurrent lines is equal to zero. On solving the determinants of the matrix which we get from the coefficient of $x,y$ from the lines given. The matrix formed using the three lines will be:$\Rightarrow \left| \begin{matrix} {{x}_{1}} & {{y}_{1}} & {{c}_{1}} \\ {{x}_{2}} & {{y}_{2}} & {{c}_{2}} \\ {{x}_{3}} & {{y}_{3}} & {{c}_{3}} \\\end{matrix} \right|=0$ The above is the formula for the coefficient of $x,y,c$ present in the line. On substituting the value we get:$\Rightarrow \left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & c \\\end{matrix} \right|=0$$\Rightarrow a\left( b\times c-a\times a \right)-b\left( b\times b-c\times a \right)+c\left( b\times a-a\times a \right)=0$$\Rightarrow a\left( bc-{{a}^{2}} \right)-b\left( {{b}^{2}}-ca \right)+c\left( ba-{{a}^{2}} \right)=0$ \[\Rightarrow 3abc-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}=0\]\[\Rightarrow {{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=0\]$\therefore $ If the lines $ax+by+c=0,bx+cy+a=0$ and $cx+ay+b=0$ be concurrent, then${{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=0$So, the correct answer is “Option C”.Note: Sometimes it may happen that the equation of the line given to us would have only a single term either of X and Y. If any of the terms will be missing then in that case the coefficient of that term will be taken as zero. For example: if a line a $x+c=0$ is given, it means that the coefficient will be written as zero in the matrix. Recently Updated PagesMaster Class 11 English: Engaging Questions & Answers for SuccessMaster Class 11 Maths: Engaging Questions & Answers for SuccessMaster Class 11 Biology: Engaging Questions & Answers for SuccessMaster Class 11 Physics: Engaging Questions & Answers for SuccessMaster Class 11 Accountancy: Engaging Questions & Answers for SuccessClass 11 Question and Answer - Your Ultimate Solutions GuideMaster Class 11 English: Engaging Questions & Answers for SuccessMaster Class 11 Maths: Engaging Questions & Answers for SuccessMaster Class 11 Biology: Engaging Questions & Answers for SuccessMaster Class 11 Physics: Engaging Questions & Answers for SuccessMaster Class 11 Accountancy: Engaging Questions & Answers for SuccessClass 11 Question and Answer - Your Ultimate Solutions Guide

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