Question

The number of elements in the relation \[ R=\{(x,y): 4x^2+y^2<52,\; x,y\in\mathbb{Z}\} \] is

• A. 67
• B. 89
• C. 86
• D. 77

Hint:

When counting integer solutions of inequalities, first restrict one variable, then count valid values of the other variable for each case.

Answer 1

The Correct Option is D

The question asks us to find the number of elements in the relation defined by:

\(R=\{(x,y): 4x^2+y^2<52,\; x,y\in\mathbb{Z}\}\)

This represents a set of integer coordinate pairs \((x, y)\) that satisfy the given inequality. Let's solve this step-by-step:

1. Understanding the Inequality:

The inequality \(4x^2 + y^2 < 52\) describes an ellipse centered at the origin with axes lengths depending on the coefficients. Here, \(4x^2\) and \(y^2\) correspond to the horizontal and vertical axes with adjustments due to the coefficients.

1. Determine Range for \(x\):

For integer values of \(x\), the possible range can be determined by:

1. \(4x^2 < 52 \Rightarrow x^2 < 13 \Rightarrow |x| < \sqrt{13} \approx 3.6\)

Since \(x\) must be an integer, \(x\) can take values from \(-3\) to \(3\).

1. Determine Possible Values for \(y\):
   • For \(x = -3\) or \(3\):
   • For \(x = -2\) or \(2\):
   • For \(x = -1\) or \(1\):
   • For \(x = 0\):
2. Totaling all the Combinations:

Now sum up all possible combinations:

• For \(x = -3\) and \(x = 3\), each with 7 values of \(y\): \(2 \times 7 = 14\)
• For \(x = -2\) and \(x = 2\), each with 11 values of \(y\): \(2 \times 11 = 22\)
• For \(x = -1\) and \(x = 1\), each with 13 values of \(y\): \(2 \times 13 = 26\)
• For \(x = 0\) with 15 values of \(y\): 15

Thus, the number of elements in the relation is 77.