SATHEE: Vectors 1 Question 29

Vectors 1 Question 29

30. Let $O A C B$ be a parallelogram with $O$ at the origin and $O C$ a diagonal. Let $D$ be the mid-point of $O A$ . Using vector methods prove that $B D$ and $C O$ intersect in the same ratio. Determine this ratio.

(1988, 3M)

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Answer:

Correct Answer: 30. ( $2 : 1$ )

Solution:

$O A C B$ is a parallelogram with $O$ as origin. Let

$\vec{O A} = \vec{a}, \vec{O B} = \vec{b}, \vec{O C} = \vec{a} + \vec{b}$

$and \vec{O D} = \frac{\vec{a}}{2}$

$\vec{C O}$ and $\vec{B D}$ meets at $P$ .

$\begin{array}{ll} ∴ & \vec{O P} = \frac{λ \cdot 0 + 1 (\vec{a} + \vec{b})}{λ + 1} \\ \Rightarrow & \vec{O P} = \frac{\vec{a} + \vec{b}}{λ + 1} \\ Again, & \vec{O P} = \frac{μ (\vec{a} \frac{2}{2}) + 1 (\vec{b})}{μ + 1} \\ \Rightarrow & \vec{O P} = \frac{μ \vec{a} + 2 \vec{b}}{2 (μ + 1)} \end{array}$

[along $\vec{O C}$ ]

[along $\vec{B D}$ ] From Eqs. (i) and (ii),

$\frac{\vec{a} + \vec{b}}{λ + 1} = \frac{μ \vec{a} + 2 \vec{a}}{2 (μ + 1)} \Rightarrow \frac{1}{λ + 1} = \frac{μ}{2 (μ + 1)}$ and $\frac{1}{λ + 1} = \frac{1}{μ + 1}$

On solving, we get $μ = λ = 2$

Thus, required ratio is $2 : 1$ .

Vectors 1 Question 29

30. Let $O A C B$ be a parallelogram with $O$ at the origin and $O C$ a diagonal. Let $D$ be the mid-point of $O A$ . Using vector methods prove that $B D$ and $C O$ intersect in the same ratio. Determine this ratio.

Answer:

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Vectors 1 Question 29

30. Let OACB be a parallelogram with O at the origin and OC a diagonal. Let D be the mid-point of OA. Using vector methods prove that BD and CO intersect in the same ratio. Determine this ratio.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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30. Let $O A C B$ be a parallelogram with $O$ at the origin and $O C$ a diagonal. Let $D$ be the mid-point of $O A$ . Using vector methods prove that $B D$ and $C O$ intersect in the same ratio. Determine this ratio.