Assertion :If a, b, c are three positive real numbers such that $a + c \neq 0$ and $\frac{1}{a} + \frac{1}{a - b} + \frac{1}{c} + \frac{1}{c - b} = 0$ then $a, b, c$ are in $H . P$ Reason: If $a, b, c$ are distinct positive real numbers such that $a (b - c) x^{2} + b (c - a) x y + c (a - b) y^{2}$ is a perfect square, then $a, b, c$ are in $H . P$ .

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
Rewrite the given expression as $\frac{1}{a} + \frac{1}{c - b} + \frac{1}{c} + \frac{1}{a - b} = 0$
$\Rightarrow \frac{a + c - b}{a (c - b)} + \frac{a + c - b}{c (a - b)} = 0$
$\Rightarrow a (c - b) = c (b - a) \Rightarrow b = \frac{2 a c}{a + c}$
For statement-2, put $t = x / y$ , and write the expression as
$y^{2} [a (b - c) t^{2} + b (c - a) t + c (a - b)]$

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Similar questions

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Reason: If 2s = (a + b + c), where a, b, c are the sides of a triangle, then area

$= \sqrt{(s - a) (s - b) (s - c)}$

Which of the following is correct?

Q. Assertion: The value of (25)³ + (−16)³ + (−9)³ is 10800.
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(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
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(d) Assertion is false and Reason is true.

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Reason: $(4, - 3)$ lies in Quadrant $I V$
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(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
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Q. Assertion: If a ray

\vec{C D}

stands on a line

\vec{A B}

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\vec{C D}

stands on a line

\vec{A B}

, then ∠ACD + ∠BCD = 180°.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Q. Assertion: If two lines AB and CD intersect at O, such that ∠AOC = 40°, then ∠BOC = 140°.

Reason: If two straight lines intersect each other, then vertically-opposite angles are equal.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

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Assertion :If a, b, c are three positive real numbers such that a+c≠0 and 1a+1a−b+1c+1c−b=0 then a,b,c are in H.P Reason: If a,b,c are distinct positive real numbers such that a(b−c)x2+b(c−a)xy+c(a−b)y2 is a perfect square, then a,b,c are in H.P.

The correct option is C Assertion is correct but Reason is incorrectRewrite the given expression as 1a+1c−b+1c+1a−b=0⇒a+c−ba(c−b)+a+c−bc(a−b)=0⇒a(c−b)=c(b−a)⇒b=2aca+cFor statement-2, put t=x/y, and write the expression asy2[a(b−c)t2+b(c−a)t+c(a−b)]