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# The diagonals of a square are along the pair of lines whose equation is $2{x^2} - 3xy - 2{y^2} = 0$. If $\left( {2,1} \right)$ is a vertex of the square, then the vertex of the square adjacent to it maybeA.$\left( {1, - 4} \right)$ B.$\left( { - 1,4} \right)$ C.$\left( {1, - 2} \right)$ D.$\left( { - 1,2} \right)$

Last updated date: 18th Sep 2024
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Hint: We will draw the plot of the square on the graph and from reference of the graph, we will substitute $x = 2$ in the given equation for the value of y – coordinate corresponding to $x = 2$. Similarly, we will calculate the x – coordinate by putting $y = 1$ in the equation. This way we will get all four vertices and then, we will see which of the options verify the answer obtained.

We are given that the diagonals of a square are along the pair of lines whose equation is $2{x^2} - 3xy - 2{y^2} = 0$.
One of the vertices of the square is given as $\left( {2,1} \right)$. We are required to find the vertex of the square adjacent to this vertex.
Now, we know that if one vertex is $\left( {2,1} \right)$ then its adjacent vertices will have their x – coordinates equal to 2 as well. Putting $x = 2$ in the equation $2{x^2} - 3xy - 2{y^2} = 0$ for the value of y – coordinate (since diagonals are along the pair of lines represented by this equation and hence vertices will also lie on this equation)., we get
$\Rightarrow 2{\left( 2 \right)^2} - 3\left( 2 \right)y - 2{y^2} = 0 \\ \Rightarrow 8 - 6y - 2{y^2} = 0 \\ \Rightarrow 2{y^2} + 6y - 8 = 0 \\ \Rightarrow 2\left( {{y^2} + 3y - 4} \right) = 0 \\ \Rightarrow {y^2} + 3y - 4 = 0 \\$
Factoring this equation, we will get the values of y as:
$\Rightarrow {y^2} + 3y - 4 = 0 \\ \Rightarrow {y^2} + 4y - y - 4 = 0 \\ \Rightarrow y\left( {y + 4} \right) - 1\left( {y + 4} \right) = 0 \\ \Rightarrow \left( {y - 1} \right)\left( {y + 4} \right) = 0 \\$
Therefore, $y = 1$ or $y = - 4$. So, the vertices will be $\left( {x,y} \right) \equiv \left( {2,1} \right){\text{ or }}\left( {2, - 4} \right)$.
Again, putting the value of $y = 1$, we get
$\Rightarrow 2{x^2} - 3x\left( 1 \right) - 2{\left( 1 \right)^2} = 0 \\ \Rightarrow 2{x^2} - 3x - 2 = 0 \\$
Factorising this equation for the value of x, we get
$\Rightarrow 2{x^2} - 4x + x - 2 = 0 \\ \Rightarrow 2x\left( {x - 2} \right) + 1\left( {x - 2} \right) = 0 \\ \Rightarrow \left( {2x + 1} \right)\left( {x - 2} \right) = 0 \\$
Therefore, $x = - \dfrac{1}{2}$ or $x = 2$. So, the vertices will be $\left( {x,y} \right) \equiv \left( { - \dfrac{1}{2},1} \right){\text{ or }}\left( {2,1} \right)$.
So, the two vertices adjacent to $\left( {2,1} \right)$ are $\left( { - \dfrac{1}{2},1} \right){\text{ and }}\left( {2, - 4} \right)$.
The graph of the square and pair of the lines can be represented as:

Here, ABCD is the square whose diagonals are along the pair of lines with equation $2{x^2} - 3xy - 2{y^2} = 0$ represented by blue coloured lines.
Now, both of these vertices don't match with any of the given options. So, we will simplify these vertices. We can shift the vertices in such a manner that they are adjacent to the vertex $\left( {2,1} \right)$ multiplying the vertex $\left( { - \dfrac{1}{2},1} \right)$ by 2 and dividing the vertex $\left( {2, - 4} \right)$ by 2. We get the vertices $\left( { - 1,2} \right)$ and $\left( {1, - 2} \right)$.
Looking at the options, we get that option (C) and option (D) are correct.

Note: In this question, be careful when we have used $x = 2{\text{ and }}y = 1$ simultaneously as we need to find the value of the other adjacent vertices. You may go wrong while plotting the graph of the square whose diagonals are along the equation of pair of lines: $2{x^2} - 3xy - 2{y^2} = 0$ and hence deducing the vertices adjacent to $\left( {2,1} \right)$.