Home
Class 12
MATHS
Let a, b,c in R and a ne 0. If alpha is ...

Let a, b,c `in` R and a `ne` 0. If `alpha` is a root of `a^(2) x^(2)` + bx + c = 0 , `beta` is a root of `a^(2) x^(2)` - bx - x = 0 and 0 `lt alpha lt beta`. Then the equation `a^(2) x^(2) + 2 bx ` + 2c = 0 has a root `gamma` that always satisfies :

Text Solution

Verified by Experts

Since `alpha` is a root of `a^(2)x^(2)+bx+c=0`.
Then `a^(2) alpha^(2)+b alpha +c=0`………i
and `beta` is a root of `a^(2)x^(2)-bx-c=0`. ltbr then `a^(2) beta^(2)-b beta-c=0`….ii
Let `f(x)=a^(2)x^(2)+2bx+2c`
`:.f(alpha)=a^(2) alpha^(2)+2b alpha+2c=a^(2)-2a^(2) alpha^(2)` [from Eq. (i)]
`=-a^(2)alpha^(2)`
`impliesf(alpha)lt0` and `f(beta)=alpha^(2)beta^(2)+2b beta+2c`
`=a^(2) beta^(2)+2a^(2) beta^(2)` [from Eq. (ii)]
`=3a^(2)beta^(2)`
`impliesf(beta)gt0`
Since `f (alpha)` and `f(beta)` are of opposite signs, then it is clear that a root `gamma` of the equation `f(x)=0` lies between `alpha` and `beta`.
Hence `alpha lt gamma lt beta [ :' alpha lt beta]`
Promotional Banner

Similar Questions

Explore conceptually related problems

If alpha and beta the roots of x^(2) + px + q = 0 and alpha^(4), beta^(4) are the roots of x^(2) - rx + s = 0 , then the equation x^(2) - 4qx + 2q^(1) - r = 0 has always :

If alpha and beta (alpha lt beta) are the roots of the equation x^(2) + bx + c = 0 , where c lt 0 lt b , then

If 0 lt a lt b lt c and the roots alpha,beta of the equation ax^2 + bx + c = 0 are non-real complex numbers, then

If alpha, beta are the rots of a x^(2)+b x+c=0 and alpha+h, beta+h are the roots of p x^(2)+q x+r=0 then h=

If a gt 0, b gt 0 and c gt 0 , then both the roots of the equations ax^2 + bx + c =0

If alpha and beta are roots of the equation x^(2) + x + 1 = 0, then alpha^(2) + beta^(2) is

If alpha and beta are the roots of x^(2) - ax + b^(2) = 0, then alpha^(2) + beta^(2) is equal to :

If the roots of the equation x^(3) + ax^(2) + bx + c = 0 are in A.P., 2a^(3) - 9ab =

Let alpha, beta be the roots of the equation x^(2) - px + r = 0 and (alpha)/(2) , 2 beta be the roots of the equation x^(2) - qx + r = 0 . Then the value of r is :

If the equation ax^(2) + 2bx - 3c = 0 has non-real roots and ((3c)/(4)) lt (a + b) , then c is always :