What is that figure?

My method isn't particularly creative, but it works.

$x = r \cos \theta \\ y = r \sin \theta$

Note that $r$ varies with $\theta$ . Plugging these into the equations and isolating $r^2$ yields:

$r^2 = \frac{1}{\frac{1 + 2 \sin \theta \cos \theta}{16} + \frac{1 - 2 \sin \theta \cos \theta}{9} }$

The area is then:

$A = \int_0^{2 \pi} \frac{1}{2} r^2 d \theta \approx 18.85$

Let $u=\frac{1}{\sqrt2}(x+y)$ and $v=\frac{1}{\sqrt2}(x-y)$ . This transformation is a rotation through $45^{\circ}$ , and doesn't affect the area of the figure.

In the new coordinate system, the equation is $\frac{u^2}{8}+\frac{2v^2}{9}=1$

This is the equation of an ellipse centred at the origin with semi-major axis $\sqrt8$ and semi-minor axis $\frac{3}{\sqrt2}$ .

Its area is just $\pi \times \sqrt8 \times \frac{3}{\sqrt2} = \boxed{6\pi}$ .

Let $x+y=4\cos α,x-y=3\sin α$

Then

$x=\dfrac {4\cos α+3\sin α}{2}$

$\implies dx=\dfrac {3\cos α-4\sin α}{2}dα$

$y=\dfrac {4\cos α-3\sin α}{2}$

So, $ydx=3-\dfrac {25}{8}\sin 2α$

Hence the required area is

$\displaystyle \int_0^{2π} \left (3-\dfrac {25}{8}\sin 2α\right ) dα$

$=6π\approx \boxed {18.849}$ .

Looking at the graph, we can tell that the equation above is in fact that of a rotated ellipse.

Writing and simplifying the expression of the above figure, we get:

$25x^2-14xy+25y^2 = 144$

The equation above is in the general form of a rotated ellipse, confirming our suspicions that it is one.

Now that we know that it is a rotated ellipse, the major axis can be found using Pythagoras' theorem, or by finding the furthest point to the origin by optimising this expression:

$d = \sqrt{x^2 + y^2}$

We can maximise or minimise to find the major and minor axis respectively.

Of course, the easier way is using Pythagoras' Theorem when shown the graph of the equation above.

We immediately see that the furthest points are $(2,2)$ and $(-2,-2)$ . The closest points are $(-1.5,1.5) ; (1.5,-1.5)$

$\sqrt{2^2 + 2^2 }$ = $\sqrt{8}$

$\sqrt{2 \times 1.5^2} = 2.121$

Plugging the values into the equation $\pi ab$ , we get $\boxed{18.85}$

Here is the solution according to my textbook
After that also I am not able to understand,
Help me to explain the solution.