Assertion -Let a- b- c be three positive real numbers which are in H-P- a-b-2a-b-c-b-2c-b- 4 Reason- If x - 0- then x-1-x- 4

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Question

Assertion :Let a, b, c be three positive real numbers which are in H.P.
$\frac{a + b}{2 a - b} + \frac{c + b}{2 c - b} \geq 4$
Reason: If $x > 0,$ then $x + \frac{1}{x} \geq 4$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is C Assertion is correct but Reason is incorrect
Assertion: As $a, b, c$ are in H.P., then
$b = \frac{2 a c}{a + c} \Rightarrow \frac{b}{a} = \frac{2 c}{a + c}$ and $\frac{b}{c} = \frac{2 a}{a + c}$
Now $\frac{a + b}{2 a - b} = \frac{1 + \frac{b}{a}}{2 - \frac{b}{a}} = \frac{a + 3 c}{2 a}$ and $\frac{c + b}{2 c - b} = \frac{1 + \frac{b}{c}}{2 - \frac{b}{c}} = \frac{3 a + c}{2 c}$
Thus, $\frac{a + b}{2 a - b} + \frac{c + b}{2 c - b}$ $\geq 1 + \frac{3}{2} (\frac{c}{a} + \frac{a}{c}) \geq 1 + \frac{3}{2} (2) = 4$
Reason: is incorrect as $x + \frac{1}{x} \geq 2$

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Assertion :Let a, b, c be three positive real numbers which are in H.P.a+b2a−b+c+b2c−b≥4 Reason: If x>0, then x+1x≥4

The correct option is C Assertion is correct but Reason is incorrectAssertion: As a,b,c are in H.P., thenb=2aca+c⇒ba=2ca+c and bc=2aa+cNow a+b2a−b=1+ba2−ba=a+3c2a and c+b2c−b=1+bc2−bc=3a+c2cThus, a+b2a−b+c+b2c−b ≥1+32(ca+ac)≥1+32(2)=4Reason: is incorrect as x+1x≥2

Assertion :Let a, b, c be three positive real numbers which are in H.P.
$\frac{a + b}{2 a - b} + \frac{c + b}{2 c - b} \geq 4$
Reason: If $x > 0,$ then $x + \frac{1}{x} \geq 4$