Question

The number of elements in the set
$S=\left\{\right(a,b):2a^2+3b^2=35;a,b\in Z\},$ where $Z$ is the set of all integers, is

Answer 1

Given set is
$S=\left\{\right(a,b):2a^2+3b^2=35;a,b\in Z\}$

As $2a^2\to \text{even number}$, so $3b^2$ has to be odd number then only the sum of them will be odd.
Possible value of
$b^2=1,9\Rightarrow b=\pm 1,\pm 3$
Putting $b^2=1$ in the given equation,
$2a^2+3=35\Rightarrow a^2=16\Rightarrow a=\pm 4$
Now, putting $b^2=9$
$2a^2+3\times 9=35\Rightarrow a^2=4\Rightarrow a=\pm 2$

The set becomes,
$S=\left\{\right(2,3),(2,–3),(–2,–3),(–2,3),(4,1),(4,–1),(–4,–1),(–4,1\left)\right\}$

Hence, the number of elements in the set is $8.$