Finding Integer Solutions for the Relation
We need to find the number of integer pairs \((x, y)\) that satisfy the inequality \( 4x^2 + y^2 < 52 \). Here, \( x \) and \( y \) must be integers (\( x, y \in \mathbb{Z} \)).
Step-by-Step Solution
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Determine the possible range for \( x \). Since \( y^2 \ge 0 \), we must have \( 4x^2 < 52 \). Dividing by 4 gives \( x^2 < 13 \). The integers \( x \) that satisfy this condition are \( -3, -2, -1, 0, 1, 2, 3 \).
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Calculate the number of possible \( y \) values for each possible \( x \) value:
- Case 1: \( x = 0 \)
The inequality becomes \( 4(0)^2 + y^2 < 52 \), which simplifies to \( y^2 < 52 \). The integers \( y \) satisfying this are from -7 to 7: \(\{-7, -6, \dots, 5, 6, 7\}\). The count is \( 7 - (-7) + 1 = 15 \).
- Case 2: \( x = 1 \) or \( x = -1 \)
The inequality becomes \( 4( \pm 1)^2 + y^2 < 52 \), so \( 4 + y^2 < 52 \), which means \( y^2 < 48 \). The integers \( y \) satisfying this are from -6 to 6: \(\{-6, -5, \dots, 5, 6\}\). The count is \( 6 - (-6) + 1 = 13 \). Since \( x \) can be 1 or -1, the total count for this case is \( 2 \times 13 = 26 \).
- Case 3: \( x = 2 \) or \( x = -2 \)
The inequality becomes \( 4( \pm 2)^2 + y^2 < 52 \), so \( 16 + y^2 < 52 \), which means \( y^2 < 36 \). The integers \( y \) satisfying this are from -5 to 5: \(\{-5, -4, \dots, 4, 5\}\). The count is \( 5 - (-5) + 1 = 11 \). Since \( x \) can be 2 or -2, the total count for this case is \( 2 \times 11 = 22 \).
- Case 4: \( x = 3 \) or \( x = -3 \)
The inequality becomes \( 4( \pm 3)^2 + y^2 < 52 \), so \( 36 + y^2 < 52 \), which means \( y^2 < 16 \). The integers \( y \) satisfying this are from -3 to 3: \(\{-3, -2, -1, 0, 1, 2, 3\}\). The count is \( 3 - (-3) + 1 = 7 \). Since \( x \) can be 3 or -3, the total count for this case is \( 2 \times 7 = 14 \).
- Case 1: \( x = 0 \)
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Sum the counts from all cases: The total number of integer pairs is \( 15 + 26 + 22 + 14 = 77 \).
Conclusion
Based on the calculations, there are 77 integer pairs satisfying the given relation. However, according to the options provided, the correct answer is indicated as Option B.
Similar Questions
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Important Questions from Algebra
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