Let \( A \) and \( B \) be two finite sets with \( m \) and \( n \) elements respectively. The total number of subsets of the set \( A \) is 56 more than the total number of subsets of \( B \). Then the distance of the point \( P(m, n) \) from the point \( Q(-2, -3) \) is:
- 10
- 6
- 4
- 8
The Correct Option is A
Approach Solution - 1
To solve the problem, we need to find the relationship between the number of elements in sets \( A \) and \( B \), and subsequently use those values to calculate the distance between the points \( P(m, n) \) and \( Q(-2, -3) \).
- The number of subsets of a set with \( k \) elements is \( 2^k \). Therefore, for set \( A \) with \( m \) elements, the number of subsets is \( 2^m \), and for set \( B \) with \( n \) elements, it is \( 2^n \).
- According to the problem, \( 2^m = 2^n + 56 \).
- Let's find suitable values of \( m \) and \( n \) that satisfy this equation:
- If we try \( 2^m = 64 \) (which is \( 2^6 \)), then \( 2^n + 56 = 64 \), so \( 2^n = 8 \) (which is \( 2^3 \)). Hence, \( m = 6 \) and \( n = 3 \).
Now, we need to find the distance between points \( P(6, 3) \) and \( Q(-2, -3) \) using the distance formula:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
- Substitute \( (x_1, y_1) = (6, 3) \) and \( (x_2, y_2) = (-2, -3) \) into the formula:
\(d = \sqrt{((-2) - 6)^2 + ((-3) - 3)^2}\) \(d = \sqrt{(-8)^2 + (-6)^2}\) \(d = \sqrt{64 + 36}\) \(d = \sqrt{100}\) \(d = 10\)
Thus, the distance between point \( P(m, n) \) and point \( Q(-2, -3) \) is 10. Therefore, the correct answer is 10.
Approach Solution -2
The total number of subsets of a set with \(m\) elements is \(2^m\) and for a set with \(n\) elements is \(2^n\). Given:
\(2^m = 2^n + 56.\)
Rearranging:
\(2^m - 2^n = 56.\)
Factoring the left side:
\(2^n (2^{m-n} - 1) = 56.\)
Since \(56 = 2^3 \times 7\), we set \(2^n = 8 \implies n = 3\) and
\(2^{m-n} - 1 = 7 \implies 2^{m-n} = 8 \implies m - n = 3.\)
Therefore:
\(m = 6, \quad n = 3.\)
The distance between points \(P(6, 3)\) and \(Q(-2, -3)\) is given by:
\(\text{Distance} = \sqrt{(6 - (-2))^2 + (3 - (-3))^2} = \sqrt{8^2 + 6^2} = \sqrt{100} = 10.\)
Thus, the correct answer is 10.
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