A linear transformation maps a point (𝑥, 𝑦) in the plane to the point (𝑥̂, 𝑦̂) according to the rule

                                  𝑥̂ = 3𝑦, 𝑦̂ = 2𝑥.

Then, the disc 𝑥 ² + 𝑦 ² ≤ 1 gets transformed to a region with an area equal to _________ . (Rounded off to two decimals)

Use π = 3.14

The correct answer is 18.84.

Step 1: Understand the Linear Transformation

We are given a linear transformation that maps any point

$(x,y)$

in the plane to a new point

$(\hat{x},\hat{y})$

according to the rules shown in the provided image:

$$\hat{x}=3y$$

$$\hat{y}=2x$$

Step 2: Express the Original Coordinates in Terms of the Transformed Coordinates

To find how the boundary of the region transforms, we express the original coordinates

$x$

and

$y$

in terms of the new coordinates

$\hat{x}$

and

$\hat{y}$

. From the transformation rules, we get:

$$y=\frac{\hat{x}}{3}$$

$$x=\frac{\hat{y}}{2}$$

Step 3: Transform the Boundary of the Disc

The original region is a disc defined by the inequality:

$$x^2+y^2\le 1$$

Substituting the expressions for

$x$

and

$y$

into the inequality yields:

$${\left(\frac{\hat{y}}{2}\right)}^2+{\left(\frac{\hat{x}}{3}\right)}^2\le 1$$

Simplifying the squared terms, we obtain the equation of an ellipse as shown in the box in the reference image:

$$\frac{{\hat{x}}^2}{3^2}+\frac{{\hat{y}}^2}{2^2}\le 1$$

Step 4: Calculate the Area of the Transformed Region

The transformed region is an ellipse with semi-axes

$a=3$

and

$b=2$

. The formula for the area of an ellipse is:

$$\text{Area}=\pi ab$$

Using the value

$\pi =3.14$

as specified in the problem, we calculate the area:

$$\text{Area}=3.14\times 3\times 2$$

$$\text{Area}=3.14\times 6=18.84$$