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If a, b, c are distinct positive real numbers and a2 + b2 + c2 = 1, then ab + bc + ca is
- a)less than 1
- b)equal to 1
- c)greater than 1
- d)any real number
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If a, b, c are distinct positive real numbers and a2 + b2 + c2 = 1, th...
Explanation:
Given Information:
a, b, c are distinct positive real numbers and a^2 + b^2 + c^2 = 1
Proof:
To find the value of ab + bc + ca, let's consider the square of (a + b + c) first:
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
(a + b + c)^2 = 1 + 2(ab + bc + ca)
Since a, b, c are distinct positive real numbers, a + b + c > 0. Therefore, we have:
(a + b + c)^2 > 0
1 + 2(ab + bc + ca) > 0
2(ab + bc + ca) > -1
ab + bc + ca > -1/2
Now, let's consider the square of (a - b - c):
(a - b - c)^2 = a^2 + b^2 + c^2 - 2(ab + bc + ca)
(a - b - c)^2 = 1 - 2(ab + bc + ca)
Since a, b, c are distinct positive real numbers, a - b - c > 0. Therefore, we have:
(a - b - c)^2 > 0
1 - 2(ab + bc + ca) > 0
-2(ab + bc + ca) > -1
ab + bc + ca < />
Combining the two inequalities, we get:
-1/2 < ab="" +="" bc="" +="" ca="" />< />
Therefore, ab + bc + ca is less than 1. Hence, the correct answer is option A.
Given Information:
a, b, c are distinct positive real numbers and a^2 + b^2 + c^2 = 1
Proof:
To find the value of ab + bc + ca, let's consider the square of (a + b + c) first:
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
(a + b + c)^2 = 1 + 2(ab + bc + ca)
Since a, b, c are distinct positive real numbers, a + b + c > 0. Therefore, we have:
(a + b + c)^2 > 0
1 + 2(ab + bc + ca) > 0
2(ab + bc + ca) > -1
ab + bc + ca > -1/2
Now, let's consider the square of (a - b - c):
(a - b - c)^2 = a^2 + b^2 + c^2 - 2(ab + bc + ca)
(a - b - c)^2 = 1 - 2(ab + bc + ca)
Since a, b, c are distinct positive real numbers, a - b - c > 0. Therefore, we have:
(a - b - c)^2 > 0
1 - 2(ab + bc + ca) > 0
-2(ab + bc + ca) > -1
ab + bc + ca < />
Combining the two inequalities, we get:
-1/2 < ab="" +="" bc="" +="" ca="" />< />
Therefore, ab + bc + ca is less than 1. Hence, the correct answer is option A.
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If a, b, c are distinct positive real numbers and a2 + b2 + c2 = 1, then ab + bc + ca isa)less than 1b)equal to 1c)greater than 1d)any real numberCorrect answer is option 'A'. Can you explain this answer?
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