Solve Question no. 4 using distance formula

​Q4. Find the value of m if the points (5 , 1) (–2, –3) and (8 , 2m ) are collinear.
 

 

Dear Student,

Please find below the solution to the asked query:

i ) We have three points  ( 5 , 1 ) , (  - 2 , - 3 ) , R( 8 , 2 m ) And these points are collinear


And we know area of triangle from given points will be zero .

We know area of triangle from given three points  :

Area  = $\frac{1}{2}\left|x_1\left(y_2 - y_{3 }\right) + x_2\left(y_3 - y_{1 }\right) + x_3\left(y_1 - y_{2 }\right)\hspace{0.17em}\right|$

Here x₁ = 5 , x₂ =  - 2 , x₃ = 8  and  y₁ = 1 , y₂ = - 3 , y₃ = 2 m

So,

$$\begin{gathered} \frac{1}{2} \left| 5\left( - 3 - 2 m\right) +\left(- 2\right)\left( 2 m - 1 \right) + 8\left( 1 - \left(- 3\right)\right)\right|\hspace{0.17em} =0 \\ \Rightarrow \frac{1}{2} \left| 5\left( - 3 - 2 m\right) - 2\left( 2 m - 1 \right) + 8\left( 1 + 3\right)\right|\hspace{0.17em} =0 \\ \Rightarrow \left| 5\left( - 3 - 2 m\right) - 2\left( 2 m - 1 \right) + 8\left( 4 \right)\right|\hspace{0.17em} =0 \\ \Rightarrow \left| - 15 - 10 m - 4 m +\hspace{0.17em}2 +32 \right|\hspace{0.17em} =0 \\ \Rightarrow \left| 19 - 14 m \right|\hspace{0.17em} =0 \\ \Rightarrow 19 - 14 m =0 \\ \Rightarrow 14 m =19 \\ \mathbf{\Rightarrow }\mathit{m}\mathbf{ }\mathbf{=}\frac{\mathbf{19}}{\mathbf{14}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)} \end{gathered}$$


Hope this information will clear your doubts about topic.

If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible.

Regards