Maths · Direction ratios and direction cosines and the angle between two intersecting lines
\boldsymbol{B}(-\hat{\boldsymbol{i}}+\boldsymbol{3} \hat{\boldsymbol{j}}+2 \hat{
\( \operatorname{Let} \boldsymbol{A}(\mathbf{2} \hat{\boldsymbol{i}}+\boldsymbol{3} \hat{\boldsymbol{j}}+\mathbf{5} \hat{\boldsymbol{k}}) \boldsymbol{B}(-\hat{\boldsymbol{i}}+\boldsymbol{3} \hat{\boldsymbol{j}}+2 \hat{\boldsymbol{k}}) \) and \( C(\lambda \hat{i}+5 \hat{j}+\mu \hat{k}) \) are vertices of \( a \) triangle and its median through \( A \) is equally inclined to the positive directions of the axes. The value of \( \lambda+ \) \( \mu \) is equal to
- A. -7
- B. 2
- C. 7
- D. 17
Step-by-step solution
The median from A goes to the midpoint D of BC. Coordinates: D = ((λ-1)/2, 4, (μ+2)/2). Vector AD = ((λ-5)/2, 1, (μ-8)/2). Being equally inclined to axes implies direction ratios equal: (λ-5)/2 = 1 = (μ-8)/2. Solving gives λ=7, μ=10, so λ+μ=17.
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