Question

Determine the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points
A (2, -2) and B (3, 7). Also find the co-ordinates of the point of division.

Answer 1

Answer:

Here is the answer according to me...

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Answer 2

$Let \: the \: line \: 2x+y-4 =0 \:divides \:the \\line \: segment \: joining \:the \: points \:A(2,-2)\\and \:B(3,7) \:in \:the \:ratio \:of \: k:1$

$Let \: intersecting \:point \:is \: D(x,y)$

$D(x,y) = \Big( \frac{k\times 3+1\times 2}{k+1} , \frac{k\times7+1\times(-2)}{k+1}\Big) \\= \Big( \frac{3k+2}{k+1}, \frac{7k-2}{k+1}\Big) \:--(1)$

$We \:know \:that,\\D(x,y) \:lies \:on \:the \: line \:2x+y-4 = 0$

$Substitute \: \Big( \frac{3k+2}{k+1}, \frac{7k-2}{k+1}\Big) \: in \: the \: Equation ,we \:get$

$\implies 2\Big(\frac{3k+2}{k+1}\Big) + \frac{7k-2}{k+1} - 4 = 0$

$\implies \frac{2(3k+2) + 7k-2 -4(k+1)}{k+1} = 0$

$\implies 6k+4 + 7k - 2 - 4k - 4 = 0$

$\implies 9k - 2 = 0$

$\implies 9k = 2$

$\implies k = \frac{2}{9}\: --(2)$

$\red { Required \:ratio (k : 1 )}\green {= 2 : 9 }$

/* Substitute Value of k in Equation (1),we get */

$( x , y ) = \Big( \frac{3\times\frac{2}{9}+2}{\frac{2}{9}+1}, \frac{7\times \frac{2}{9}-2}{\frac{2}{9}+1}\Big)\\= \Big( \frac{\frac{6+18}{9}}{\frac{2+9}{9}} , \frac{\frac{14-18}{9}}{\frac{2+9}{9}}\Big) \\= \Big( \frac{24}{11} , \frac{-4}{11}\Big)$

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