Question

S is the disk given by\begin{aligned}{u^2} + {v^2}1;\; \hfill \\x = au,y = bv \hfill \\ \end{aligned}

Short answer

Therefore, Image of S is the region inside an ellipse

Step-by-step solution

Define the term ellipse.

The locus of all those locations in a plane whose sum of distances from two fixed points in the plane is constant is called an ellipse. The foci (singular focus) are the fixed points that are encircled by the curve. The constant ratio is the eccentricity of the ellipse, and the fixed line is the directrix. Eccentricity is a property of the ellipse that indicates its elongation and is symbolised by the letter 'e.'

Obtain the image.

Given that

\({u^2} + {v^2} \le 1\)

Substitute the expressions for \(u\)and \(v\)

\(\begin{array}{l}x = au \to u = \frac{x}{a}\\y = bv \to v = \frac{y}{b}\end{array}\)

\({\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} \le 1\)

Therefore Image of S is the region inside an ellipse