Consider the following statements: 1. If x is directly proportional to z and y is directly proportional to z, then (x 2- y 2) is directly proportional to z 2. 2. If x is inversely proportional to z and y is inversely proportional to z, then (xy) is inversely proportional to z 2. Which of the above statements is/are correct?
Both 1 and 2
Analyzing Proportionality Statements
We are asked to examine two statements involving direct and inverse proportionality between variables x, y, and z.
Statement 1 Analysis: Direct Proportionality
The first statement says:
"If x is directly proportional to z and y is directly proportional to z, then (\(x^2\) - \(y^2\)) is directly proportional to \(z^2\)."
Let's represent the given proportionality relationships mathematically:
- x is directly proportional to z means \(x \propto z\). This can be written as \(x = k_1 z\), where \(k_1\) is a non-zero constant of proportionality.
- y is directly proportional to z means \(y \propto z\). This can be written as \(y = k_2 z\), where \(k_2\) is a non-zero constant of proportionality.
Now let's consider the expression (\(x^2\) - \(y^2\)) and substitute the expressions for x and y:
\((x^2 - y^2) = (k_1 z)^2 - (k_2 z)^2\)
\((x^2 - y^2) = k_1^2 z^2 - k_2^2 z^2\)
\((x^2 - y^2) = (k_1^2 - k_2^2) z^2\)
Let \(K = k_1^2 - k_2^2\). Since \(k_1\) and \(k_2\) are constants, \(K\) is also a constant. Thus, we have:
\((x^2 - y^2) = K z^2\)
This equation shows that (\(x^2\) - \(y^2\)) is equal to a constant times \(z^2\). This is the definition of direct proportionality. Therefore, (\(x^2\) - \(y^2\)) is directly proportional to \(z^2\). However, we must consider the case where \(K = 0\). If \(k_1^2 - k_2^2 = 0\), then \(k_1^2 = k_2^2\), meaning \(k_1 = \pm k_2\). If \(k_1 = k_2\) (and both are non-zero), then \(K=0\), and \(x=y\). In this case, \(x^2-y^2 = 0\), which is \(0 \cdot z^2\). While technically proportional with a constant 0, in typical definitions of proportionality, the constant is non-zero. If \(k_1 = -k_2\) (and both are non-zero), \(K = k_1^2 - (-k_1)^2 = k_1^2 - k_1^2 = 0\). So \(x^2-y^2 = 0\) again. If either \(k_1\) or \(k_2\) is zero, the premise (\(x \propto z\) or \(y \propto z\)) fails unless x or y is identically zero. Assuming \(x\) and \(y\) are not identically zero, \(k_1\) and \(k_2\) are non-zero. The proportionality holds even if \(K=0\). Thus, statement 1 is correct.
Statement 2 Analysis: Inverse Proportionality
The second statement says:
"If x is inversely proportional to z and y is inversely proportional to z, then (xy) is inversely proportional to \(z^2\)."
Let's represent the given inverse proportionality relationships mathematically:
- x is inversely proportional to z means \(x \propto \frac{1}{z}\). This can be written as \(x = \frac{k_3}{z}\), where \(k_3\) is a non-zero constant of proportionality.
- y is inversely proportional to z means \(y \propto \frac{1}{z}\). This can be written as \(y = \frac{k_4}{z}\), where \(k_4\) is a non-zero constant of proportionality.
Now let's consider the expression (xy) and substitute the expressions for x and y:
\((xy) = \left(\frac{k_3}{z}\right) \left(\frac{k_4}{z}\right)\)
\((xy) = \frac{k_3 k_4}{z^2}\)
Let \(M = k_3 k_4\). Since \(k_3\) and \(k_4\) are non-zero constants, \(M\) is also a non-zero constant. Thus, we have:
\((xy) = \frac{M}{z^2}\)
This equation shows that (xy) is equal to a non-zero constant times \(\frac{1}{z^2}\). This is the definition of inverse proportionality to \(z^2\). Therefore, (xy) is inversely proportional to \(z^2\). Statement 2 is correct.
Conclusion
Based on our analysis, both Statement 1 and Statement 2 are correct.
| Statement | Given Relationship(s) | Derived Relationship | Correctness |
|---|---|---|---|
| 1 | \(x \propto z\), \(y \propto z\) | \((x^2 - y^2) \propto z^2\) | Correct |
| 2 | \(x \propto \frac{1}{z}\), \(y \propto \frac{1}{z}\) | \((xy) \propto \frac{1}{z^2}\) | Correct |
Revision Table: Proportionality Concepts
| Concept | Mathematical Notation | Explanation |
|---|---|---|
| Direct Proportionality | \(a \propto b\) or \(a = kb\) | As 'b' increases, 'a' increases proportionally (if k > 0). The ratio \(\frac{a}{b}\) is constant. |
| Inverse Proportionality | \(a \propto \frac{1}{b}\) or \(a = \frac{k}{b}\) | As 'b' increases, 'a' decreases proportionally (if k > 0). The product \(a \cdot b\) is constant. |
Additional Information: Working with Proportionality
When dealing with proportionality, it's useful to remember the following:
- Proportionality can be converted into an equation by introducing a constant of proportionality. For example, if \(A \propto B\), then \(A = k B\).
- Operations on proportional quantities can result in new proportional relationships. For instance, if \(x \propto z\), then \(x^n \propto z^n\) for any power n (provided expressions are defined).
- If \(x \propto z\) and \(y \propto z\), then \((x+y) \propto z\) and \((ax+by) \propto z\) for any constants a and b.
- If \(x \propto z\) and \(y \propto z\), then \(\frac{x}{y} = \frac{k_1 z}{k_2 z} = \frac{k_1}{k_2}\), which is a constant (assuming \(y \ne 0\)). This means x is proportional to y, and y is proportional to x.
- If \(x \propto \frac{1}{z}\), then \(x^2 \propto \left(\frac{1}{z}\right)^2 = \frac{1}{z^2}\).
These properties help simplify the analysis of relationships between variables when they are proportional.
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