The line 2x + y −4 = 0 divides the line segment joining

Question:

The line 2x + y −4 = 0 divides the line segment joining A(2, −2) and B(3, 7) in the ratio

(a) 2 : 5
(b) 2 : 9
(c) 2 : 7
(d) 2 : 3

 

Solution:

(b) 2 : 9

Let the line $2 x+y-4=0$ divide the line segment in the ratio $k: 1$ at the point $P$.

Then, by section formula, the coordinates of P are

$P\left(\frac{3 k+2}{k+1}, \frac{7 k-2}{k+1}\right)$

Since $P$ lies on the line $2 x+y-4=0$, we have:

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

$\Rightarrow(6 k+4)+(7 k-2)-(4 k+4)=0$

$\Rightarrow 9 k=2$

$\Rightarrow k=\frac{2}{9}$

Hence, the required ratio is $\frac{2}{9}: 1$, which is same as $2: 9$.

 

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