SATHEE: Vectors 1 Question 28

Vectors 1 Question 28

29. In a $△ O A B, E$ is the mid-point of $B O$ and $D$ is a point on $A B$ such that $A D : D B = 2 : 1$ . If $O D$ and $A E$ intersect at $P$ , determine the ratio $O P : P D$ using methods.

$(1989, 4 M)$

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Answer:

Correct Answer: 29. ( $3 : 2$ )

Let $O$ be origin and $\vec{O A} = \vec{a}, \vec{O B} = \vec{b}$

$\vec{O E} = \frac{\vec{b}}{2}$

[since $E$ being mid-point of $\vec{O B}$ ]

$\vec{O D} = \frac{\vec{a} \cdot 1 + \vec{b} \cdot 2}{1 + 2}$

(since, $D$ divides $\vec{A B}$ in the ratio of $2 : 1$ )

$\Rightarrow$ Equation of $\vec{O D}$ is $\vec{r} = t (\frac{\vec{a} + 2 \vec{b}}{3})$

and equation of $\vec{A E}$ is $\vec{r} = \vec{a} + s (\frac{\vec{b}}{2} - \vec{a})$

If $\vec{O D}$ and $\vec{A E}$ intersect at $P$ , then there must be some $\vec{r}$ for which they are equal.

$\begin{aligned} \Rightarrow t (\frac{\vec{a} + 2 \vec{b}}{3}) = \vec{a} + s (\frac{\vec{b}}{2} - \vec{a}) \\ \Rightarrow \frac{t}{3} = 1 - s and \frac{2 t}{3} = \frac{s}{2} \\ \Rightarrow t = \frac{3}{5} and s = \frac{4}{5} \\ ∴ Point P is \frac{\vec{a} + 2 \vec{b}}{5} . \end{aligned}$

Since, $P$ divides $\vec{O D}$ in the ratio of $λ : 1$ .

$∴ \frac{λ (\frac{\vec{a} + 2 \vec{b}}{3}) + 1 \cdot 0}{λ + 1} = (\frac{\vec{a} + 2 \vec{b}}{5})$

From Eqs. (i) and (ii),

$\begin{aligned} \frac{λ}{3 (λ + 1)} & = \frac{1}{5} \\ \Rightarrow & 5 λ & = 3 λ + 3 \\ \Rightarrow & λ & = \frac{3}{2} \\ ∴ & \frac{O P}{P D} & = \frac{3}{2} \end{aligned}$

Vectors 1 Question 28

29. In a $△ O A B, E$ is the mid-point of $B O$ and $D$ is a point on $A B$ such that $A D : D B = 2 : 1$ . If $O D$ and $A E$ intersect at $P$ , determine the ratio $O P : P D$ using methods.

Answer:

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Vectors 1 Question 28

29. In a △OAB,E is the mid-point of BO and D is a point on AB such that AD:DB=2:1. If OD and AE intersect at P, determine the ratio OP:PD using methods.

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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29. In a $△ O A B, E$ is the mid-point of $B O$ and $D$ is a point on $A B$ such that $A D : D B = 2 : 1$ . If $O D$ and $A E$ intersect at $P$ , determine the ratio $O P : P D$ using methods.