For the points \( (a,0) \), \( (0,b) \) and \( (1,1) \), Which of the following statements is/are TRUE?
(A) The 3 points are collinear if \( a+b=ab \)
(B) The 3 points are collinear if \( a+b\ne ab \)
(C) The 3 points are non-collinear if \( a+b=ab \)
(D) The 3 points are non-collinear if \( a+b\ne ab \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (D) only
Three points are collinear exactly when the area of the triangle they form is zero:
\( \displaystyle \frac{1}{2}\left|a(b-1)+0\cdot(1-0)+1\cdot(0-b)\right|=0 \)
\( \displaystyle ab-a-b=0 \)
\( \displaystyle a+b=ab \)
So the points are collinear when \( a+b=ab \), and non-collinear when \( a+b\ne ab \).
Checking the statements
(A) The 3 points are collinear if \( a+b=ab \) — Correct. Then the area \( \frac{1}{2}|ab-a-b| \) is zero.
(B) The 3 points are collinear if \( a+b\ne ab \) — Incorrect. Then \( ab-a-b\ne 0 \), so the area is not zero.
(C) The 3 points are non-collinear if \( a+b=ab \) — Incorrect. Then the area is zero, so the points are collinear.
(D) The 3 points are non-collinear if \( a+b\ne ab \) — Correct. Then the area \( \frac{1}{2}|ab-a-b| \) is not zero.