Question

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.

Answer 1

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) $(x_2,y_2)$

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

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

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

$4y-4=2x-10$

$2x-4y=6$

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

Substituting (11,4) in (1)

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

$\Rightarrow$ 11 - 8 = 3 

$\Rightarrow$ 3 = 3 

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

Now, put x = 0 in equation 1

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

Now, put y= 0 in equation 1

$\Rightarrow$ x = 3 

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

So, this is your answer.

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