If a - b - c are distinct positive real numbers such that b a - c -2 ac then the roots of ax 2- bx - c -0 areA- real and equalB- real and distinctC- ImaginaryD- data insufficient

The correct option is C Imaginary b=2ac/a+c ⇒ b is HM of a amp; c. ⇒ ax^2 + 2bx + c = 0 ∴ Δ = 4b^2 4ac =4(b^2 ac) GM of a amp; c is √(ac) ∴ ac gt;b^ ...

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Standard XIII

Mathematics

Discriminant

If a , b , c ...

Question

If a, b, c are distinct positive real numbers such that b(a + c) = 2ac then the roots of $a x^{2} + b x + c = 0$ are

real and equal

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real and distinct

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Imaginary

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data insufficient

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Solution

The correct option is C
Imaginary

$b = \frac{2 a c}{a + c} \Rightarrow$ b is HM of a & c.
$\Rightarrow a x^{2} + 2 b x + c = 0 ∴ Δ = 4 b^{2} - 4 a c = 4 (b^{2} - a c)$
GM of a & c is $\sqrt{a c}$
$∴ a c > b^{2} (G M \geq H M) ∴ Δ < 0$

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If a, b, c are distinct positive real numbers such that b(a + c) = 2ac then the roots of ax2+bx+c=0 are

The correct option is C Imaginary b=2aca+c ⇒ b is HM of a & c. ⇒ax2+2bx+c=0∴ Δ=4b2−4ac=4(b2−ac) GM of a & c is √ac ∴ ac>b2 (GM≥HM)∴ Δ<0

If a, b, c are distinct positive real numbers such that b(a + c) = 2ac then the roots of $a x^{2} + b x + c = 0$ are

The correct option is C
Imaginary

$b = \frac{2 a c}{a + c} \Rightarrow$ b is HM of a & c.
$\Rightarrow a x^{2} + 2 b x + c = 0 ∴ Δ = 4 b^{2} - 4 a c = 4 (b^{2} - a c)$
GM of a & c is $\sqrt{a c}$
$∴ a c > b^{2} (G M \geq H M) ∴ Δ < 0$