SATHEE: Vectors 5 Question 12

Vectors 5 Question 12

12. Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (you may assume that the trapezium is not a parallelogram).

(1998, 8M)

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Solution:

Let $O$ be the origin of reference. Let the position vectors of $A$ and $B$ be $\vec{a}$ and $\vec{b}$ respectively.

Since, $B C | O A, \vec{B C} = α \vec{O A} = α \vec{a}$ for some constant $α$ .

Equation of $O C$ is $\vec{r} = t (\vec{b} + α \vec{a})$ and

equation of $A B$ is $\vec{r} = \vec{a} + λ (\vec{b} - \vec{a})$

Let $P$ be the point of intersection of $O C$ and $A B$ . Then, at point $P, t (\vec{b} + α \vec{a}) = \vec{a} + λ (\vec{b} - \vec{a})$ for some values of $t$ and $λ$ .

$\Rightarrow (t α - 1 + λ) \vec{a} = (λ - t) \vec{b}$

Since, $\vec{a}$ and $\vec{b}$ are non-parallel vectors, we must have

$t α - 1 + λ = 0 and λ = t \Rightarrow t = 1 / (α + 1)$

Thus, position vector of $P$ is $\vec{r_{1}} = \frac{1}{α + 1} (\vec{b} + α \vec{a})$

Equation of $M N$ is $\vec{r} = \frac{1}{2} \vec{a} + k [\vec{b} + \frac{1}{2} (α - 1) \vec{a}]$

For $k = \frac{1}{α + 1}$ , which is the coefficient of $\vec{b}$ in $\vec{r_{1}}$ , we get

$\vec{r} = \frac{1}{2} \vec{a} + \frac{1}{α + 1} [\vec{b} + \frac{1}{2} (α - 1) \vec{a}]$

$= \frac{1}{(α + 1)} \vec{b} + \frac{1}{2} (α - 1) \cdot \frac{1}{α + 1} \vec{a} + \frac{1}{2} \vec{a}$

$= \frac{1}{(α + 1)} \vec{b} + \frac{1}{2 (α + 1)} (α - 1 + α + 1) \vec{a}$

$= \frac{1}{(α + 1)} (\vec{b} + α \vec{a}) = {\vec{r}}_{1} \Rightarrow P$ lies on $M N$ .

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अगला

Vectors 5 Question 12

12. Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (you may assume that the trapezium is not a parallelogram).

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Vectors 5 Question 12

12. Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (you may assume that the trapezium is not a parallelogram).

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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