Value of $m$ for which points are collinear.

Question

Find value of $m$ for which $(1,4),(4,5)$ and $(m,m)$ are collinear.

What I try : If points are collinear , Then all points lie on same line

So equation of line through $(1,4)$ and $(4,5)$ is

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

$\displaystyle 3(y-4)=x-1 $

Now point $(m,m)$ also lie on that line.

So we have $\displaystyle 3(m-4)=m-1\Longrightarrow m=\frac{11}{2}$

But answer is $\displaystyle m=\frac{7}{2}$

Please explain me where i am wrong

Thanks

Answer 1

I think your answer is correct, but I can give an alternate way for your problem. Assume the three given points to be coordinates of a triangle . Now , if the points lie on one line i.e, collinear, then it is only possible if the area of the triangle tends to zero ... So with this assumption you can find the area of this triangle with determinant method and then set the area, which will be a linear equation with variable m to be zero then you can get your result .I hope it will be helpful.

Answer 2

your answer is correct:

If the points $A(1,4), B(5,1)$ and $C(m,m)$ are collinear, they are on the same line having same gradient(slope). $$m_{AB}=m_{BC}=m_{AC}=\frac{y_2-y_1}{x_2-x_1}$$ $$\frac{5-4}{4-1}=\frac{m-4}{m-1}$$

$$m=\frac{11}{2}$$