If points A(60 i +3 j ), B(40 i -8 j ) and C(a i +52 j ) are collinear, then a is equal to,

If points A(60 i +3 j ), B(40 i -8 j ) and C(a i +52 j ) are coll…

MCQ

If points $\text{A}(60\hat{\text{i}}+3\hat{\text{j}}),\ \text{B}(40\hat{\text{i}}-8\hat{\text{j}})$ and $\text{C}(\text{a}\hat{\text{i}}+52\hat{\text{j}})$ are collinear, then a is equal to,

A
$40$
✓
$-40$
C
$20$
D
$-20$

✓

Answer

Correct option: B.

$-40$

Given: Three points $\text{A}(60\hat{\text{i}}+3\hat{\text{j}}),\ \text{B}(40\hat{\text{i}}-8\hat{\text{j}})$ and $\text{C}(\text{a}\hat{\text{i}}+52\hat{\text{j}})$ are collinear.
Then, $\overrightarrow{\text{AB}}=\lambda\overrightarrow{\text{BC}}$
We have, $\overrightarrow{\text{AB}}=\big(40\hat{\text{i}}-8\hat{\text{j}}\big)-\big(60\hat{\text{i}}+3\hat{\text{j}}\big)$
$=-20\hat{\text{i}}-11\hat{\text{j}}$
$\overrightarrow{\text{BC}}=\big(\text{a}\hat{\text{i}}-52\hat{\text{j}}\big)-\big(40\hat{\text{i}}-8\hat{\text{j}}\big)$
$=(\text{a}-40)\hat{\text{i}}-44\hat{\text{j}}$
Therefore, $\overrightarrow{\text{AB}}=\lambda\overrightarrow{\text{BC}}$
$\Rightarrow-20\hat{\text{i}}-11\hat{\text{j}}=\lambda(\text{a}-40)\hat{\text{i}}-\lambda44\hat{\text{j}}$
$\Rightarrow\lambda(\text{a}-40)=-20,\ -44\lambda=-11\Rightarrow\lambda=\frac{1}4$
$\Rightarrow\text{a}-40=-80$
$\Rightarrow\text{a}=-40$

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