If the sum of 3 non-zero distinct real numbers a, b and c is 2 , and the two sets {a,b,c} and {1/a ,1/b ,1/c} are the same, what is the value of a^2 +b^2 +c^2 ?
To find the value of a^2 + b^2 + c^2, we need to first determine the values of a, b, and c.
Given that the sum of the three numbers is 2, we can set up the equation:
a + b + c = 2 ...(Equation 1)
Since the two sets {a, b, c} and {1/a, 1/b, 1/c} are the same, it means that each element in one set has a reciprocal in the other set.
We can express this as:
1/a + 1/b + 1/c = a + b + c ...(Equation 2)
Multiplying both sides of Equation 2 by abc will eliminate the denominators:
bc + ac + ab = a^2bc + ab^2c + abc^2
Rearranging this equation gives us:
a^2bc + ab^2c + abc^2 - bc - ac - ab = 0
Factoring out the common term "abc" gives us:
abc(a + b + c - 1) = 0
Since the equation holds true for any values of a, b, and c, we have two possible cases to consider:
Case 1: abc = 0
This case implies that at least one of the numbers a, b, or c is zero. However, we are given that the numbers are non-zero, so this case is not possible.
Case 2: a + b + c - 1 = 0
Rearranging this equation gives us:
a + b + c = 1
Now, we have a system of two equations (Equation 1 and a + b + c = 1) with two variables (a, b, and c) that we can solve simultaneously.
Subtracting Equation 1 from a + b + c = 1, we get:
(a + b + c) - (a + b + c) = 1 - 2
0 = -1
This implies that there is no solution to the system of equations a + b + c = 1 and a + b + c = 2.
Therefore, there is no unique value for a, b, and c that satisfies both conditions, and as a result, the value of a^2 + b^2 + c^2 is not determinable.
To find the value of a^2 + b^2 + c^2, we need to examine the given information. Let's start by analyzing the condition that the sum of the three distinct real numbers a, b, and c is 2.
We can express this condition as the equation: a + b + c = 2.
Next, we are told that the sets {a, b, c} and {1/a, 1/b, 1/c} are the same. This implies that each element in one set has a reciprocal counterpart in the other set. Mathematically, this can be represented as:
1/a = b or 1/a = c
1/b = a or 1/b = c
1/c = a or 1/c = b
Now, we can solve these equations to find the values of a, b, and c.
From the first equation, we have two possibilities:
Case 1: 1/a = b
Substituting this into the equation a + b + c = 2, we get:
a + 1/a + c = 2
Similarly, from the second equation, we find:
b + 1/b + c = 2
Since the sets {a, b, c} and {1/a, 1/b, 1/c} are the same, we can equate these two equations:
a + 1/a + c = b + 1/b + c
This simplifies to:
a + 1/a = b + 1/b
To solve this equation, we can multiply both sides by ab to eliminate the denominators:
a^2 + 1 = ab + b
Rearranging the terms, we obtain:
a^2 - ab + 1 - b = 0
a^2 - (a - 1)b + 1 = 0
This is a quadratic equation in terms of a, with coefficients (-b), (a - 1), and 1. To find the solutions for a, we can apply the quadratic formula:
a = [-(a - 1) ± √((a - 1)^2 - 4(1)(1))] / (2)
a = [1 - a ± √(a^2 - 2a + 1 - 4)] / 2
a = (1 - a ± √(a^2 - 2a - 3)) / 2
Now, we can analyze the two cases of solutions for a.
Case 1: a = (1 - a + √(a^2 - 2a - 3)) / 2
Simplifying this equation, we get:
a = 1/2
Plugging this value into the equation a + b + c = 2, we find:
1/2 + b + c = 2
Rearranging the terms, we obtain:
b + c = 3/2
Therefore, b and c must be distinct real numbers that sum up to 3/2.
Case 2: a = (1 - a - √(a^2 - 2a - 3)) / 2
Simplifying this equation, we get:
a = -3
However, when we substitute this value into the equation a + b + c = 2, we get:
-3 + b + c = 2
b + c = 5
Since we are given that a, b, and c are distinct real numbers, we can conclude that the second case is not valid. Therefore, a = 1/2.
Now, we can find the values of b and c using the equation a + b + c = 2:
1/2 + b + c = 2
Rearranging the terms, we find:
b + c = 3/2
Since we have an equation with two variables and one equation, we can determine the values of b and c by setting one of the variables to a constant and solving for the other variable. For simplicity, let's set b = 3/4.
Using this value, we can find c by substitution:
(3/4) + c = 3/2
c = 3/2 - 3/4
c = 6/4 - 3/4
c = 3/4
Therefore, the values of a, b, and c are: a = 1/2, b = 3/4, c = 3/4.
Finally, we can calculate the value of a^2 + b^2 + c^2:
(1/2)^2 + (3/4)^2 + (3/4)^2 = 1/4 + 9/16 + 9/16 = 5/8 + 9/16 = 10/16 + 9/16 = 19/16
Hence, the value of a^2 + b^2 + c^2 is 19/16.