Question

Find the relation between $x$ and $y$ if the points $A\left( {x,y} \right),B\left( { - 5,7} \right)$ and $C\left( { - 4,5} \right)$ are collinear.

Answer 1

We need to find the relation between $x$ and $y$ as all the given points are collinear. We know that the points $A,B$ and $C$ are collinear then area of $\Delta ABC = 0$. So, we will calculate area of triangle using $A = \dfrac{1}{2}\left[ {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]$ and put it equal to 0 to find the relation between $x$ and $y$.

Complete step by step solution: We will first consider the given points $A\left( {x,y} \right),B\left( { - 5,7} \right)$ and $C\left( { - 4,5} \right)$.
We need to find the relation between $x$ and $y$ if the given points are collinear.
Now, we know that the points $A\left( {x,y} \right),B\left( { - 5,7} \right)$ and $C\left( { - 4,5} \right)$ are collinear then area of $\Delta ABC = 0$.
Thus, we will use $A = \dfrac{1}{2}\left[ {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]$ to find the area of triangle where $\left( {{x_1},{y_1}} \right);\left( {{x_2},{y_2}} \right)$ and $\left( {{x_3},{y_3}} \right)$ are the coordinated of $A,B$ and $C$.
Hence, we will substitute the value of coordinates given in the formula and find the value of area of triangle,
Thus, we get,

$$\Rightarrow A = \dfrac{1}{2}\left[ {x\left( {7 - 5} \right) - 5\left( {5 - y} \right) - 4\left( {y - 7} \right)} \right] \\\ \Rightarrow A = \dfrac{1}{2}\left[ {2x - 25 + 5y - 4y + 28} \right] \\\ \Rightarrow A = \dfrac{1}{2}\left[ {2x + y + 3} \right] \\\$$

Now, as we know that the points $A,B$ and $C$ are collinear then area of $\Delta ABC = 0$.
Thus, we get,

$$\Rightarrow \dfrac{1}{2}\left[ {2x + y + 3} \right] = 0 \\\ \Rightarrow 2x + y + 3 = 0 \\\ \Rightarrow 2x + y = - 3 \\\$$

Hence, we get the relation between $x$ and $y$ is $2x + y = - 3$.

Note: We must remember that the coordinates are collinear when the area of the triangle is equal to zero and using this fact only, we have found the relation between $x$ and $y$. We have to remember the formula, $A = \dfrac{1}{2}\left[ {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]$ to determine the area of triangle using the coordinates of the triangle. We can easily determine the relation as one of the coordinates are given as $\left( {x,y} \right)$.