Class 12 Determinants · Long Answer3 Markstriangle_area
Class 12 Maths · Determinants · Long Answer #92
Prove the points A(a,b+c),B(b,c+a),C(c,a+b) are collinear.
✅ Step-by-Step Solution
Final Answer: The three points are collinear.
- The collinearity determinant is D=\(\begin{vmatrix}a & b+c & 1\\b & c+a & 1\\c & a+b & 1\end{vmatrix}\).
- Let s=a+b+c. In every row, the first coordinate plus the second equals s, so column 1 plus column 2 equals s times column 3.
- The columns are linearly dependent, hence D=0. Thus the triangle area |D|/2=0.
- Therefore the three points lie on the line x+y=a+b+c (including any coincident-point cases).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The collinearity determinant is D=\(\begin{vmatrix}a & b+c & 1\\b & c+a & 1\\c & a+b & 1\end{vmatrix}\).
- [1 mark] Let s=a+b+c. In every row, the first coordinate plus the second equals s, so column 1 plus column 2 equals s times column 3.
- [1 mark] The columns are linearly dependent, hence D=0. Thus the triangle area |D|/2=0.
- [0.5 marks] Therefore the three points lie on the line x+y=a+b+c (including any coincident-point cases).
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