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Prove that the points whose coordinates are respectively (5,1),(1,-1), and (11,4) lie on a straight line, and find its intercepts on the axis.

Last updated date: 26th Mar 2023
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Hint: In this type of question first find out the equation of the straight line using two points then satisfy the third point in this equation, then put x = 0, and y = 0 you will get your intercepts.

Given: Equation of line joining (5,1) $({x_1},{y_1})$ and (1, - 1) $({x_2},{y_2})$

$y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\left( {x - {x_1}} \right) \\$

$y - 1 = \dfrac{{ - 1 - 1}}{{1 - 5}}\left( {x - 5} \right) \\$

$y - 1 = \dfrac{2}{4}\left( {x - 5} \right) \\$

$4y - 4 = 2x - 10 \\$

$2x - 4y = 6 \\$

$x - 2y = 3...............................\left( 1 \right) \\$

Substituting (11,4) in (1)

$\Rightarrow$ (11) - 2 x 4 = 3

$\Rightarrow$ 11 - 8 = 3

$\Rightarrow$ 3 = 3

$\Leftrightarrow$ It satisfies the equation of line, so all the points lie on the same straight line.

Now, put x = 0 in equation 1

$\Rightarrow y = - \dfrac{6}{4} = - \dfrac{3}{2}$

Now, put y= 0 in equation 1

$\Rightarrow$ x = 3

So, x intercept is 3 and y intercept is -$\dfrac{3}{2}$

NOTE: - Here $\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ is nothing but the slope of the line joining two points $(x_1, y_1)$ and $(x_2, y_2)$.