Question

If \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\), then \(x y+y z+z x\) is equal to : (a) \(\frac{(a+b+c)^{2}}{x^{2}+y^{2}+z^{2}}\) (b) \(\frac{x^{2}(a+b+c)^{2}-a^{2}\left(x^{2}+y^{2}+z^{2}\right)}{2 a^{2}}\) (c) \(\frac{a x+b y+c z}{(a+b+c)^{2}}\) (d) none of the above

Short answer

Question: Given \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\), find the expression for \(xy+yz+zx\), where the options are: (a) \(\frac{(a+b+c)^{2}}{x^{2}+y^{2}+z^{2}}\) (b) \(\frac{x^{2}(a+b+c)^{2}-a^{2}\left(x^{2}+y^{2}+z^{2}\right)}{2 a^{2}}\) (c) \(\frac{a x+b y+c z}{(a+b+c)^{2}}\) (d) none of the above Answer: (c)

Step-by-step solution

Make use of given condition

We are given that \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\). Let us assume that the common ratio is '\(k\)'. Then, we have \(x = a k\), \(y = b k\), and \(z = c k\).

Find expression for xy+yz+zx

Using the values of x, y, and z in terms of 'k', we can find the expression for 'xy+yz+zx'. Here is the calculation: \(xy + yz + zx = (a k)(b k) + (b k)(c k) + (c k)(a k) = k^2(ab+bc+ca)\).

Compare with given options

Now we will compare this expression with the given options: (a) \(\frac{(a+b+c)^{2}}{x^{2}+y^{2}+z^{2}} = \frac{(a+b+c)^{2}}{(ak)^{2}+(bk)^{2}+(ck)^{2}} = \frac{(a+b+c)^{2}}{k^2(a^2+b^2+c^2)}\) (b) \(\frac{x^{2}(a+b+c)^{2}-a^{2}\left(x^{2}+y^{2}+z^{2}\right)}{2 a^{2}} = \frac{(ak)^{2}(a+b+c)^{2}-a^{2}[(ak)^{2}+(bk)^{2}+(ck)^{2}]}{2 a^{2}}\) (c) \(\frac{a x+b y+c z}{(a+b+c)^{2}} = \frac{a(ak)+b(bk)+c(ck)}{(a+b+c)^{2}} = \frac{k(ab+bc+ca)}{(a+b+c)^{2}}\) (d) none of the above Comparing the found expression for \(xy+yz+zx\) with these options, we can see that it matches with option (c) after multiplying the denominator and numerator by \(k\) in the expression. So, the correct answer is: \(xy+yz+zx = \boxed{\frac{a x+b y+c z}{(a+b+c)^{2}}}\).

Key concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that contain numbers, variables, and operators to express a value or a relationship between variables. For example, in the given problem, the equation \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\) is a set of algebraic expressions that are assumed to be equal to each other, setting up a proportional relationship.

Working with algebraic expressions involves understanding how to manipulate and combine these entities following algebraic rules. In our problem, we were able to express \(x\), \(y\), and \(z\) in terms of a constant \(k\) and the respective variable coefficients \(a\), \(b\), and \(c\). This transformation into \(ak\), \(bk\), and \(ck\) simplifies the problem and lets us find the required expression for \(xy+yz+zx\) effectively.

Ratio and Proportion

The concept of ratio and proportion is a fundamental aspect of quantitative aptitude, crucial in comparing quantities. A ratio is a way to compare two quantities by using division, while a proportion states that two ratios are equivalent.

In the exercise, the relationship \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\) is a proportion involving three ratios. It is critical to interpret this correctly, as it indicates that for some constant \(k\), the variables \(x\), \(y\), and \(z\) are directly proportional to \(a\), \(b\), and \(c\) respectively. Understanding this relationship is key in solving the problem and is a valuable skill in various areas such as mixing solutions, comparing growth rates, and dealing with financial ratios.

Problem-Solving Strategies

Effective problem-solving strategies are essential in mathematics to navigate and solve complex questions. A structured approach can simplify the process and lead to a solution more efficiently.

In our solution, we followed a step-by-step strategy:

• First, we identified the given conditions and what was asked for.
• Next, we assumed a common ratio \(k\), linking the variables with the constants, reducing the complexity of the expressions.
• Then, we derived a simplified expression for \(xy+yz+zx\) using the values of \(x\), \(y\), and \(z\) in terms of \(k\).
• Finally, we compared the derived expression with the given options to find the correct answer.

Strategies like these guide students in breaking down problems into manageable parts and applying algebraic principles strategically to arrive at a solution.