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Question

What is the distance between the points which divide the line segment joining $$(4,3)$$ and $$(5,7)$$ internally and externally in the ratio $$2:3$$?


A
12175
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B
13175
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C
175
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D
6175
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Solution

The correct option is A $$\cfrac { 12\sqrt { 17 } }{ 5 } $$
Given $$(4, 3)$$ and $$(5, 7)$$
Internally and externally in ratio
(1) Internal division:
$$x = \dfrac {2(5) + 3(4)}{5}, y = \dfrac {2(7) + 3(3)}{2 + 3} \rightarrow (x, y) = \left (\dfrac {22}{5}, \dfrac {23}{5}\right )$$

(2) External division:
$$x = \dfrac {2(5) - 3(4)}{2 - 3}, y = \dfrac {2(7) - 3(3)}{2 - 3} \rightarrow (x, y) = (2, -5)$$

$$Distance = \sqrt {\left (\dfrac {2{2}}{5} -( 2)\right )^{2} + \left (\dfrac {23}{5} - (5)\right )^{2}} = \sqrt {\dfrac {144}{25} + \dfrac {2304}{25}} = \dfrac {12\sqrt {17}}{5}$$.

Mathematics

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