The position vectors of point A, B and C are \( \lambda\hat{\imath}+3\hat{\jmath} \), \( 12\hat{\imath}+\mu\hat{\jmath} \) and \( 11\hat{\imath}-3\hat{\jmath} \) respectively. If C divides the line segment joining A and B in the ratio 3 : 1, then the values of \( \lambda \) and \( \mu \) are :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \lambda=8,\mu=-5 \)
By the section formula, the point dividing AB internally in the ratio m : n = 3 : 1 has position vector
\( \displaystyle \vec{c}=\frac{m\vec{b}+n\vec{a}}{m+n}=\frac{3\vec{b}+\vec{a}}{4} \)
\( \displaystyle 11\hat{\imath}-3\hat{\jmath}=\frac{3(12\hat{\imath}+\mu\hat{\jmath})+(\lambda\hat{\imath}+3\hat{\jmath})}{4} \)
\( \displaystyle =\frac{(36+\lambda)\hat{\imath}+(3\mu+3)\hat{\jmath}}{4} \)
Comparing the components:
\( \displaystyle 36+\lambda=44\ \Rightarrow\ \lambda=8 \)
\( \displaystyle 3\mu+3=-12\ \Rightarrow\ \mu=-5 \)
Tip: In the section formula the ratio number of B multiplies \( \vec{b} \) (the far end); a quick check is that C with ratio 3 : 1 must lie closer to B.