Question

Acute angle through which co-ordinate axes are rotated in order to remove \[xy\] term in \[5{x^2} + 2xy + 3{y^2} + 4x + 5y + 1 = 0\] is \[\dfrac{\pi }{n}\] where (2n) equals

Answer 1

Hint: In this question we will use the concept of rotation of axes to find the angle of rotation in order to remove the \[xy\] term from the given equation. For that, we will use the formula that came from the general term, and for finding the angle of rotation we will use the trigonometric relations. With the help of the formula, we will find n and by substituting n in 2n we get the required answer.

Formula Used:
When the equation is given as \[A{x^2} + Bxy + C{y^2} + Dx + Ey + F = 0\]
Then the angle of rotation is calculated in order to remove the \[xy\] term will be calculated using the cotangent formula \[\cot 2\theta = \dfrac{{A - C}}{B}\]

Complete step by step answer:
In this, we will be putting the values of A, B, and C in order to find the angle to remove \[xy\] term by comparing the given equation with the general equation.
Now, let us compare the given equation \[5{x^2} + 2xy + 3{y^2} + 4x + 5y + 1 = 0\] with the general equation \[A{x^2} + Bxy + C{y^2} + Dx + Ey + F = 0\]
By comparing the like terms in both the equations we get,
A=5, B=2, and C=3
We know that the angle of rotation is found using the given cotangent formula \[\cot 2\theta = \dfrac{{A - C}}{B}\]
Let us now substitute the values of A, B, and C in the formula we get,
\[\Rightarrow \cot 2\theta = \dfrac{{5 - 3}}{2}\]
By subtracting 3 from 5 we get,
\[\Rightarrow \cot 2\theta = \dfrac{2}{2}\]
On dividing the terms on the left-hand side of the above equation we get,
\[\Rightarrow \cot 2\theta = 1\]
We know that using trigonometric relations
\[\Rightarrow \cot \dfrac{\pi }{4} = 1\]
Now let us compare \[\cot 2\theta = 1\] with the known trigonometric relation we get,
\[\Rightarrow 2\theta = \dfrac{\pi }{4}\]
Now on dividing both sides by 2 we get,
\[\Rightarrow \theta = \dfrac{\pi }{8}\]
So, the angle of rotation for which the “\[xy\] ” term should be removed is \[\theta = \dfrac{\pi }{8}\]
Now we will be equating the angle with the result given in the problem, \[\dfrac{\pi }{n}\]
Let us now compare both angles
\[\Rightarrow \dfrac{\pi }{n} = \dfrac{\pi }{8}\]
By cross multiplying and solving we get,
\[\Rightarrow n = 8\]
Here we have to find the value of (2n),
Let us put the value of n in (2n) we get
\[\Rightarrow 2n = 2 \times 8\]
\[\Rightarrow 2n = 16\]

Hence the value of (2n) is 16.

Note:
Here while finding the value of n we are in need of trigonometric relation of cotangent. While finding the angle of rotation we use the cotangent formula because the angle of rotation is done along the acute angle.