If $a, b, c$ are rational number $(a > b > c > 0)$ and quadratic equation $(a + b - 2 c) x^{2} + (b + c - 2 a) x + (c + a - 2 b) = 0$ has a root in the interval $(- 1, 0)$ , then which of the following statement(s) is/are correct?

a x^{2} + 2 b x + c = 0

has both roots negative

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Both roots are rational

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

c x^{2} + 2 b x + a = 0

has both roots negative

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

a + c < 2 b

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct option is D $a + c < 2 b$
The given equation has one root as 1.
$1. α = \frac{c + a - 2 b}{a + b - 2 c}$
So, the other root is $x = \frac{c + a - 2 b}{a + b - 2 c}$
As $\frac{c + a - 2 b}{a + b - 2 c} < 0 \Rightarrow a + c < 2 b$
For equation $a x^{2} + 2 b x + c = 0$
$f (0) = c > 0, \frac{- 2 b}{a} < 0$
So both roots negative
Similarly option (D) is true

Suggest Corrections

Similar questions

Q. If

a, b, c

are rational number

(a > b > c > 0)

and quadratic equation

(a + b - 2 c) x^{2} + (b + c - 2 a) x + (c + a - 2 b) = 0

has a root in the interval

(- 1, 0)

, then which of the following statement(s) is/are correct?

Q. If

a, b, c

are distinct rational numbers, then the roots of the quadratic equation

(a + b - 2 c) x^{2} + (b + c - 2 a) x + (c + a - 2 b) = 0

are

If a, b, c are positive rational numbers such that a > b > c and the quadratic equation
$(a + b - 2 c) x^{2} + (b + c - 2 a) x + (c + a - 2 b) = 0$ has a root in the interval (-1, 0), then

Q. If

a, b, c

are distinct rational numbers, then the roots of the quadratic equation

(a + b - 2 c) x^{2} + (b + c - 2 a) x + (c + a - 2 b) = 0

are

Q. If

a, b, c

are distinct rational numbers, then the roots of the quadratic equation

(a + b - 2 c) x^{2} + (b + c - 2 a) x + (c + a - 2 b) = 0

are

Join BYJU'S Learning Program

If a, b, c are rational number (a>b>c>0) and quadratic equation (a+b−2c)x2+(b+c−2a)x+(c+a−2b)=0 has a root in the interval (−1,0), then which of the following statement(s) is/are correct?

The correct option is D a+c<2bThe given equation has one root as 1. 1.α=c+a−2ba+b−2c So, the other root is x=c+a−2ba+b−2c As c+a−2ba+b−2c<0⇒a+c<2b For equation ax2+2bx+c=0 f(0)=c>0,−2ba<0 So both roots negative Similarly option (D) is true

If $a, b, c$ are rational number $(a > b > c > 0)$ and quadratic equation $(a + b - 2 c) x^{2} + (b + c - 2 a) x + (c + a - 2 b) = 0$ has a root in the interval $(- 1, 0)$ , then which of the following statement(s) is/are correct?