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Home  >>  CBSE XI  >>  Math  >>  Straight Lines
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In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?

$\begin {array} {1 1} (A)\;2:1 & \quad (B)\;1:2 \\ (C)\;1:3 & \quad (D)\;3:1 \end {array}$

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1 Answer

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Toolbox:
  • Equation of a line joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is $ \large\frac{y-y_1}{y_2-y_1}$$ = \large\frac{x-x_1}{x_2-x_1}$
  • Section formula : $\large\frac{mx_2+nx_1}{m+n}$$, \large\frac{my_2+ny_1}{m+n}$
The given points are (-1, 1) and (5,7).
Hence equation of the line joining the above points is
$ \large\frac{y-1}{7-1}$$ = \large\frac{x-(-1)}{5-(-1)}$
$ \Rightarrow \large\frac{y-1}{6}$$ = \large\frac{x+1}{6}$
$ \Rightarrow y-1 = x+1$
or $x-y+2=0$---------(1)
Equation of the given line is
$x+y-4=0$---------(2)
The point of intersection of the lines (1) and (2) is
$\qquad x-y=-2$
$ \qquad x+y=4$
$ \qquad 2x \quad = 2$
$ \Rightarrow x = 1$ and $y = 3$
Let the point (1,3) divide the line segment joining (-1,1) and (5,7) in the ratio 1 : k
By applying the section formula,
$ 1 = \large\frac{k(-1)+1(5)}{k+1}$
$ \Rightarrow k +1 = -k+5$
$ \Rightarrow 2k=4$
or $ k = 2$
Hence the line joining the points (-1, 1) and (5,7) is divided by line
$ x+y=4$ in the ratio 1 : 2
answered May 21, 2014 by thanvigandhi_1
 

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