Question:
If \(8\vec{AB}-3\vec{AC}-5\vec{AD}=\vec{0}\), then the point \(C\) divides \(BD\) in the ratio ...
If \(8\vec{AB}-3\vec{AC}-5\vec{AD}=\vec{0}\), then the point \(C\) divides \(BD\) in the ratio ...
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Take A as the origin, express C in terms of B and D and compare with the section formula.
Updated On: Nov 12, 2026
- \(8:5\) internally
- \(3:4\) externally
- \(4:3\) internally
- \(5:8\) externally
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
Take A as the origin, so \(\vec{AB}=\vec{b}\), \(\vec{AC}=\vec{c}\), \(\vec{AD}=\vec{d}\).
Step 2: Solve for C
\(8\vec{b}-3\vec{c}-5\vec{d}=\vec{0}\Rightarrow\vec{c}=\frac{8\vec{b}-5\vec{d}}{3}\).
Step 3: Compare with the section formula
For a point dividing BD in the ratio \(m:n\), \(\vec{c}=\frac{m\vec{d}+n\vec{b}}{m+n}\). Here \(m=-5\), \(n=8\), and \(m+n=3\).
Step 4: Interpret
The ratio is \(BC:CD=-5:8\). A negative ratio means external division, so C divides BD externally in the ratio \(5:8\).
Final Answer:
C divides BD externally in the ratio 5 : 8. \[ \boxed{\text{(D) }5:8\text{ externally}} \]
Take A as the origin, so \(\vec{AB}=\vec{b}\), \(\vec{AC}=\vec{c}\), \(\vec{AD}=\vec{d}\).
Step 2: Solve for C
\(8\vec{b}-3\vec{c}-5\vec{d}=\vec{0}\Rightarrow\vec{c}=\frac{8\vec{b}-5\vec{d}}{3}\).
Step 3: Compare with the section formula
For a point dividing BD in the ratio \(m:n\), \(\vec{c}=\frac{m\vec{d}+n\vec{b}}{m+n}\). Here \(m=-5\), \(n=8\), and \(m+n=3\).
Step 4: Interpret
The ratio is \(BC:CD=-5:8\). A negative ratio means external division, so C divides BD externally in the ratio \(5:8\).
Final Answer:
C divides BD externally in the ratio 5 : 8. \[ \boxed{\text{(D) }5:8\text{ externally}} \]
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