MCQ
If the points $A(1, 2), O(0, 0) $ and $C(a, b)$ are collinear then :
- A$a = b$
- B$a = mb$
- ✓$2a = b$
- D$a + b = 0$
✓
Answer
Correct option: C.
$2a = b$
The given points are $A (1,2), O(0,0)$ and $C(a, b)$ are collinear.
$\therefore(\text{x}_1=1,\text{y}_1=2),(\text{x}_2=0,\text{y}_2=0)$ and $(\text{x}_1=\text{a},\text{y}_1=\text{b})$
The given points are collinear.
$\Rightarrow\text{x}_1(\text{y}_2-\text{y}_3)+\text{x}_2(\text{y}_3-\text{y}_1)+\text{x}_3(\text{y}_1-\text{y}_2)=0$
$\Rightarrow1(0-\text{b})+0+\text{a}(2-0)=0$
$\Rightarrow-\text{b}+2\text{a}=0$
$\Rightarrow2\text{a}=\text{b}$
$\therefore(\text{x}_1=1,\text{y}_1=2),(\text{x}_2=0,\text{y}_2=0)$ and $(\text{x}_1=\text{a},\text{y}_1=\text{b})$
The given points are collinear.
$\Rightarrow\text{x}_1(\text{y}_2-\text{y}_3)+\text{x}_2(\text{y}_3-\text{y}_1)+\text{x}_3(\text{y}_1-\text{y}_2)=0$
$\Rightarrow1(0-\text{b})+0+\text{a}(2-0)=0$
$\Rightarrow-\text{b}+2\text{a}=0$
$\Rightarrow2\text{a}=\text{b}$
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