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If the points $(3,-2)$, $(x,2)$ and $(8,8)$are collinear, then find the value of $10x$ using determinant.

Last updated date: 20th Jun 2024
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Hint: In order to solve this question we have to know about collinearity and it’s condition. For this question as we have to use determinant we can use the area condition that for collinear points the area enclosed is zero. Apply this condition using the determinant and get the final expression and calculate the value of $x$ and multiply it with 10 and it will be the answer.

Complete step by step answer:
Before solving the question let’s know about collinearity.
Collinear means lying on the same line which means that if three points are collinear then they must lie on the same line.
Now, it is specifically written that we have to use determinants in order to solve this question.
As we know that collinear points lie on the same line which means that if we join all these three points then the closed figure formed will have area equal to zero because figure will be straight line.
We know that, if $A({{x}_{1}},{{y}_{1}})$, $B({{x}_{2}},{{y}_{2}})$ and $C({{x}_{3}},{{y}_{3}})$ then the area formed after joining these three points with each other using determinant is equal to $\dfrac{1}{2}\times \left| \begin{matrix}
   {{x}_{1}} & {{y}_{1}} & 1 \\
   {{x}_{2}} & {{y}_{2}} & 1 \\
   {{x}_{3}} & {{y}_{3}} & 1 \\
\end{matrix} \right|$.
Extending this concept in the question, the points $(3,-2)$, $(x,2)$ and $(8,8)$are collinear means that the area formed after joining these points with each other, then area = 0.
$\therefore \dfrac{1}{2}\times \left| \begin{matrix}
   3 & -2 & 1 \\
   x & 2 & 1 \\
   8 & 8 & 1 \\
\end{matrix} \right|=0$
Expanding the determinant along row 1, we get
$\Rightarrow \dfrac{1}{2}\left\{ 3\times (2-8)-(-2)(x-8)+1\times (8x-16) \right\}=0$
After simplifying the above equation, we get
$\Rightarrow (-18+2x-16+8x-16)=0$
$\Rightarrow 10x-50=0$
$\Rightarrow x=\dfrac{50}{10}=5$
Hence, the value of $x$ is equal to 5.
 So, $10x=10\times 5=50$

Hence, the required answer is 50.

Note: This question is also simple but the tricky part is to use determinants to solve it. There are three different conditions of collinearity as there is slope condition, distance condition and area condition but only the area condition involves the use of determinants. As it is specifically written that we have to use determinants but students often make mistakes by applying other two conditions. Hence, carefully read the question and apply conditions according to it and get the correct answer.