Question

Show that the points are collinear $ ( - 5,1), (5,5), (10,7) $

Answer 1

Hint:
Here we must know that collinearity of three points means that they all lie on the same line. So with the help of two points we can make the equation of the line joining those two points. Then if the third point has to lie on it then that point must satisfy the given equation of the line. Hence in this way we can check its collinearity.

Complete step by step solution:
Here we are told to show that the given three points are collinear. So we need to know what the meaning of collinearity of three points is. This means that all the three points lie on the same line. So the simplest way to solve this problem is that we can find the equation of the line using two points. We know that we can find the equation of the line if we have two points given because we can find their slope.
Let us suppose we take the two points $ ( - 5,1), (5,5) $

We know that if we have two points $ (a,b),(c,d) $ then the slope of the line joining these two points is $ \dfrac{{d - b}}{{c - a}} $
So we have two points $ ( - 5,1),(5,5) $
So we can say that slope $ = m = \dfrac{{5 - 1}}{{5 - ( - 5)}} = \dfrac{4}{{5 + 5}} = \dfrac{4}{{10}} = \dfrac{2}{5} $
Hence we get $ m = \dfrac{2}{5} $
Now we have got the slope of the line joining the two points $ ( - 5,1),(5,5) $
We know that if we have any point say $ (a,b) $ and the slope $ m $ then we can find the equation of the line by the formula
 $ (y - b) = m(x - a) $
Here we have the slope and two points. We can take any point to find the equation of the line. Let us take the point $ ( - 5,1) $
So we will get
 $
  (y - 1) = \dfrac{2}{5}(x - ( - 5)) \\
  5(y - 1) = 2(x + 5) \\
  5y - 5 = 2x + 10 \\
  2x - 5y + 15 = 0 \\
  $
Hence we get that the equation of the line joining the two points $ ( - 5,1),(5,5) $ is $ 2x - 5y + 15 = 0 $
Now we can check if the third point which is $ (10,7) $ lies on this line or not. If it lies on this line then we can say that all the points are collinear. Hence let us put $ x = 10,y = 7 $ in the above equation of the line we got.
 $ 2x - 5y + 15 = 0 $
 $
  2(10) - 5(7) + 15 = 0 \\
  20 - 35 + 15 = 0 \\
   - 15 + 15 = 0 \\
  0 = 0 \\
  $
Which is true.
Hence we get that all the three points are collinear.
Hence proved.

Note:
Here we can also solve this by the alternate method where we assume that these three points are the points of the triangle. But for their collinearity the area formed by the joining of all these three points must be zero as all these points lie on the straight line.