Determine whether the given points are collinear. (2) P(1, 2) , Q(2, 8/5 ) , R(3,6/5 )
2. Determine whether the given points are collinear.
(2) P(1, 2) , Q(2, 8/5 ) , R(3,6/5 )
\[P\left( 1, 2 \right), Q\left( 2, \frac{8}{5} \right), R\left( 3, \frac{6}{5} \right)\]
Slope of PQ = \[\frac{\frac{8}{5} - 2}{2 - 1} = \frac{\frac{- 2}{5}}{1} = \frac{- 2}{5}\]
Slope of QR = \[\frac{\frac{6}{5} - \frac{8}{5}}{3 - 2} = \frac{\frac{- 2}{5}}{1} = \frac{- 2}{5}\]
So, the slope of PQ = slope of QR.
Point Q lies on both the lines.
Hence, the given points are collinear.
Explanation:-
The given problem involves finding if the given points are collinear or not. The given points are:
[P\left( 1, 2 \right), Q\left( 2, \frac{8}{5} \right), R\left( 3, \frac{6}{5} \right)]
To determine if the points are collinear, we need to check if the slope of PQ is equal to the slope of QR.
Slope of PQ can be calculated using the formula:
[m_{PQ} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}]
Substituting the given values in the formula, we get:
[m_{PQ} = \frac{\frac{8}{5} – 2}{2 – 1} = \frac{\frac{- 2}{5}}{1} = \frac{- 2}{5}]
Similarly, slope of QR can be calculated as:
[m_{QR} = \frac{\frac{6}{5} – \frac{8}{5}}{3 – 2} = \frac{\frac{- 2}{5}}{1} = \frac{- 2}{5}]
As we can see, the slope of PQ is equal to the slope of QR. Also, point Q lies on both the lines. Therefore, we can conclude that the given points are collinear.
Chapter 5. Co-ordinate Geometry – Problem Set 5 (Page 122)