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Question

Find the ratio in which the line segment joining the points $$(-3, 10)$$ and $$(6, -8)$$ is divided by $$(-1, 6)$$


A
1:7
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B
2:7
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C
2:5
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D
3:7
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Solution

The correct option is B $$2:7$$
Using the section formula, if a point $$(x,y)$$ divides the line joining the points $$({ x }_{ 1 },{ y }_{ 1 })$$ and $$\displaystyle ({ x }_{ 2 },{ y }_{ 2 })$$ in the ratio $$
m:n $$, then $$(x,y) = \left( \dfrac { m{ x }_{ 2 } + n{ x }_{ 1 } }{ m + n }
,\dfrac { m{ y }_{ 2 }  + n{ y }_{ 1 } }{ m + n }  \right) $$
Let the ratio be $$ k : 1 $$

Substituting $$({ x }_{ 1 },{ y }_{
1 }) = (-3,10) $$ and $$({x }_{ 2 },{ y }_{ 2 }) = (6,-8) $$  in the section
formula, we get  $$\displaystyle \left( \frac { k(6)  + 1(-3) }{ k + 1 }
,\frac { k(-8) + 1(10) }{ k + 1 }  \right) = ( -1, 6) $$ 

$$\displaystyle \left( \frac { 6k - 3 }{ k + 1 } ,\frac { -8k + 10 }{ k + 1} \right) = ( -1,6)$$

Comparing the x - coordinate,

$$ \displaystyle \frac { 6k-3 }{ k+1 } =-1 $$
$$ \implies  6k - 3 = -k - 1 $$

$$ 7k = 2 $$

$$\displaystyle  k = \frac {2}{7} $$

Hence, the ratio is $$ 2:7$$


232558_239050_ans_42053defbb10488fb445fdd5034b3080.png

Mathematics

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