Question

The number of elements in the set {$(a,b):2a^2+3b^2=35$, $a,b$ belongs to $Z$}, where $Z$ is the set of all integers, is

• A. $2$
• B. $4$
• C. $8$
• D. $12$
• E. $16$

Answer 1

The correct option is C

$8$

Explanation for the correct option:

Step 1: Find all the possible values of $b$.

An equation $2a^2+3b^2=35$ is given, where $x,y$ are integers.

Rewrite the given equation as follows:

$3b^2=35-2a^2$

Since the maximum value of the right-hand side will be $35$.

so,

$$\begin{gathered} 3b^2\le 35 \\ \Rightarrow b^2\le 11.6 \end{gathered}$$

Therefore, the possible values of $b$ are$0,\pm 1,\pm 2,\pm 3,\pm 4$.

Since equation $3b^2=35-2a^2$ is given.

We know that multiple of $2$ is an even number and $35$ is an odd number and the difference between an odd and even number is always an odd number.

Therefore, $3b^2$ is an odd number.

For $b=0$:

$3b^2=0$, Since $0$ is neither an odd nor even number, therefore, $b=0$ is rejected.

For $b=\pm 1$:

$3b^2=3$, Since $3$ is an odd number, therefore, $b=\pm 1$ is accepted.

For $b=\pm 2$:

$3b^2=12$, Since $12$ is an even number, therefore, $b=\pm 2$ is rejected.

For $b=\pm 3$:

$3b^2=27$, Since $12$ is an even number, therefore, $b=\pm 3$ is accepted.

Therefore, the possible values of $b$ are $\pm 1,\pm 3$.

Step 2: Find the number of elements in the given set.

Compute the values of $a$ corresponding to the values of $b$.

For $b=\pm 1$.

$$\begin{gathered} 3{(\pm 1)}^2=35-2a^2 \\ \Rightarrow 3=35-2a^2 \\ \Rightarrow 2a^2=35-3 \\ \Rightarrow 2a^2=32 \\ \Rightarrow a^2=16 \\ \Rightarrow a=\pm 4 \end{gathered}$$

Therefore, for $b=\pm 1$ $a=\pm 4$.

For $b=\pm 3$.

$$\begin{gathered} 3{(\pm 3)}^2=35-2a^2 \\ \Rightarrow 27=35-2a^2 \\ \Rightarrow 2a^2=35-27 \\ \Rightarrow 2a^2=8 \\ \Rightarrow a^2=4 \\ \Rightarrow a=\pm 2 \end{gathered}$$

Therefore, for $b=\pm 3$ $a=\pm 2$.

So, the elements in the required set are $\left\{\right(1,4),(-1,4),(1,-4),(-1,-4),(2,3\left)\right(2,-3),(-2,3),(-2,-3\left)\right\}$.

Therefore, there are $8$ elements in the required set.

Hence, option $C$ is the correct answer.