Question

In each of following examples find the co-ordinates of point A which divides segmentPQ in ration a:b
1)P(-3,4) Q(1,-4)a:b=2
2)P(-2,-5)Q(4,3)a:b=3:4
3)P(2,6)Q(-4,1)a:b=1:2

Answer 1

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Formula used here
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$\bf \underline{co - ordinates} \\ \\ = > x = \frac{mx _{2} + nx _{1} }{m + n} \\ \\ = > y = \frac{m y_{2} + ny _{1} }{m + n} \\ \\ \bf \: where \: \: m :n = ratio \: \: of \: \: division$
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For

1)
Ratio = 2:1 =m:n
$= > x = \frac{ (1 \times 2) + (- 3 \times 1)}{2 + 1} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: = \frac{2 - 3}{3} = \bf \underline{- \frac{1}{3} } \\ \\ = > y = \frac{( - 4 \times2 ) + ( 4\times1) }{2 + 1} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: = \frac{ - 8 + 4}{3} = \bf \underline{ - \frac{4}{3} }$

So, A =( - ⅓, - 4/3)

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2)
Ratio = 3:4

$= > x = \frac{ ( 4 \times 3) + ( - 2 \times 4)}{3+4} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: = \frac{ 12 - 8}{7} = \bf \underline{ \frac{4}{7} } \\ \ \\ = > y = \frac{( 3 \times3 ) + ( - 5\times4) }{3 + 4} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: = \frac{ 9 - 2 0 }{7} = \bf \underline{ - \frac{11}{7} }$

So, A = (4/7, - 11/7)
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3)

Ratio = 1:2

$= > x = \frac{ ( - 4 \times1 ) + ( 2\times 2)}{ 1 + 2} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: = \frac{ - 4 + 4}{3} = \bf \underline{0} \\ \\ = > y = \frac{( 1 \times 1 ) + ( 6\times2) }{ 1 + 2} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: = \frac{ 1 + 12}{3} = \bf \underline{ \frac{13}{3} }$
So, A = (0, 13/3)

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