Question

The mean of \[a,b\] and \[c\] is \[x\]. If \[ab + bc + ca = 0\], what is the meaning of \[{a^2},{b^2}\] and \[{c^2}\]?
A) \[\dfrac{{{x^2}}}{3}\]
B) \[{x^2}\]
C) \[3{x^2}\]
D) \[9{x^2}\]

Answer 1

Hint:
Here we will find the relation between the given terms and their meanings. Then we will find the mean of \[{a^2},{b^2}\] and \[{c^2}\] by adding and subtracting suitable terms such that the expression resembles some identity. Then we will apply suitable algebraic identity and simplify the expression. We will substitute the given value as well as obtained relation to get the required answer.

Formula Used:
Mean \[ = \] Sum of all the terms \[ \div \] Number of terms

Complete step by step solution:
The mean of \[a,b\] and \[c\] is given as \[x\]:
So, we can write it as
Mean \[ = \dfrac{{a + b + c}}{3}\]
\[ \Rightarrow x = \dfrac{{a + b + c}}{3}\]
Multiplying 3 on both sides, we get
\[ \Rightarrow 3x = a + b + c\]
So we get one value as \[a + b + c = 3x\]……\[\left( 1 \right)\]
We also have value of \[ab + bc + ca = 0\]…..\[\left( 2 \right)\]
Now, we have to find the mean of \[{a^2},{b^2}\] and \[{c^2}\]
Mean \[ = \dfrac{{{a^2} + {b^2} + {c^2}}}{3}\]
Now we will make the numerator a square of \[a + b + c\]. For that, we will add and subtract \[2ab,2bc\] and \[2ca\] in the numerator and get,
\[ \Rightarrow \] Mean \[ = \dfrac{{{a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca - 2ab - 2bc - 2ca}}{3}\]
\[ \Rightarrow \] Mean \[ = \dfrac{{\left( {{a^2} + {b^{}} + {c^2} + 2ab + 2bc + 2ca} \right) - \left( {2ab + 2bc + 2ca} \right)}}{3}\]
Now using the algebraic identity \[{\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca\], we get
\[ \Rightarrow \] Mean \[ = \dfrac{{{{\left( {a + b + c} \right)}^2} - 2\left( {ab + bc + ca} \right)}}{3}\]
Now substituting value from equation \[\left( 1 \right)\] and \[\left( 2 \right)\] in above equation, we get
\[ \Rightarrow \] Mean \[ = \dfrac{{{{\left( {3x} \right)}^2} - 2 \times 0}}{3}\]
Simplifying the expression, we get
\[ \Rightarrow \] Mean \[ = \dfrac{{9{x^2}}}{3} = 3{x^2}\]
So, we get the mean of \[{a^2},{b^2}\] and \[{c^2}\] as \[3{x^2}\].

Hence, option (C) is correct.

Note:
Mean of numbers is defined as the sum of all the terms of an observation divided by the numbers of terms of that observation. Mean is usually termed as an average of all the terms. We know that there are three types of mean and they are arithmetic mean, geometric mean, and harmonic mean. We should not get confused between the mean and median of a data set. Mean is the central tendency of the data set but the median is the central value of the data set.