Question

Two sets $A$ and $B$ are as under :
$A=\left\{\right(a,b)\in R\times R:|a-5|<1\text{and}|b-5|<1\};$
$B=\left\{\right(a,b)\in R\times R:4(a-6{)}^2+9(b-5{)}^2\le 36\}.$ Then

• A. neither $A\subset B$ nor $B\subset A$
• B. $B\subset A$
• C. $A\subset B$
• D. $A\cap B=\varphi$ (an empty set)

Answer 1

The correct option is C $A\subset B$

Given:
$A=\left\{\right(a,b)\in R\times R:|a-5|<1\text{and}|b-5|<1\};$
Since $|a-5|<1$
$\therefore -1<a-5<1$
$\Rightarrow a\in (4,6)$
Similarly $b\in (4,6)$
and $B=\left\{\right(a,b)\in R\times R:4(a-6{)}^2+9(b-5{)}^2\le 36\}$
Since $4(a-6{)}^2+9(b-5{)}^2\le 36$
$\therefore \frac{(a-6{)}^2}{9}+\frac{(b-5{)}^2}{4}\le 1$

From diagram, it is clear that $A\subset B$