If A (a, a − b), B (a + b, − b) and C (b, a) are three collinear points, where \( a,b\in\mathrm{R} \), then :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( a^{2}-b^{2}=ab \)
Three points are collinear exactly when the area of the triangle formed by them is zero:
\( \displaystyle \frac{1}{2}\begin{vmatrix} a & a-b & 1\\ a+b & -b & 1\\ b & a & 1\end{vmatrix}=0 \)
Expanded, this is the same as
\( x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})=0 \)
With the three given points:
\( a(-b-a)+(a+b)\left(a-(a-b)\right)+b\left((a-b)-(-b)\right)=0 \)
\( -ab-a^{2}+(a+b)b+ab=0 \)
\( -ab-a^{2}+ab+b^{2}+ab=0 \)
\( -a^{2}+b^{2}+ab=0 \)
\( a^{2}-b^{2}=ab \)