If - are the roots of ax2 - bx - c - 0 and - are the roots of a1x2 - b1x -c1 - 0- then - are roots of Ax2 - x - C - 0- where C-1 -

The correct option is D bc+b_1c_1 Using the formula for sum and product of roots, C=βγ(γ +β ) c_1b_1=αγ(αγ ) cb=βα(α+ β ) we can see that ...

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Standard XII

Mathematics

Discriminant

If α,β are ...

Question

If $α, β$ are the roots of $a x^{2} + b x + c = 0$ and $- α, γ$ are the roots of $a_{1} x^{2} + b_{1} x + c_{1} = 0$ , then $β, γ$ are roots of $A x^{2} + x + C = 0$ , where $C^{- 1} =$

\frac{c}{b} + \frac{c_{1}}{b_{1}} = 0

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\frac{c}{a} + \frac{c_{1}}{a_{1}}

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\frac{a}{c} + \frac{a_{1}}{c_{1}}

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\frac{b}{c} + \frac{b_{1}}{c_{1}}

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Solution

The correct option is D $\frac{b}{c} + \frac{b_{1}}{c_{1}}$
Using the formula for sum and product of roots,
$- C = \frac{β γ}{(γ + β)}$
$\frac{c_{1}}{b_{1}} = \frac{α γ}{(α γ)}$
$\frac{- c}{b} = \frac{β α}{(α + β)}$
we can see that $\frac{1}{C}$ = $\frac{b}{c} + \frac{(b_{1})}{(c_{1})}$

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Similar questions

Q. If

α, β

are the roots of

a x^{2} + b x + c = 0, α_{1}, - β

are the roots of

a_{1} x^{2} + b_{1} x + c_{1} = 0

, show that

α, α_{1}

are the roots of

\frac{x^{2}}{\frac{b}{a} + \frac{b_{1}}{a_{1}}} + x + \frac{1}{\frac{b}{c} + \frac{b_{1}}{c_{1}}} = 0

Q. lf

f (x) = x^{3} + a x^{2} + b x + c = 0

has roots

a, b, c

and

a, b, c \in R

. If the roots of

x^{3} + a_{1} x^{2} + b_{1} x + c_{1} = 0

are

(α - β)^{2}, (β - γ)^{2}

and

(γ - α)^{2}

, then for

c_{1} = 0

, roots of

f (x) = 0

are

Q. Assertion :If

\frac{a}{a_{1}}, \frac{b}{b_{1}}, \frac{c}{c_{1}}

are in

A . P

then

a_{1}, b_{1}, c_{1}

are in

G . P

Reason: If

a x^{2} + b x + c

and

a_{1} x^{2} + b_{1} x^{2} + c_{1} = 0

have a common root &

\frac{a}{a_{1}}, \frac{b}{b_{1}}, \frac{c}{c_{1}}

are in

A . P

then

a_{1}, b_{1}, c_{1}

are in

G . P

Q. If

f (x) = a_{1} x^{2} + b_{1} x + c_{1}

and

α, β

are the roots of

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

. Find

f (α) f (β)

Q. Let

α, β

be the roots of

a x^{2} + b x + c = 0, a \neq 0

and

α_{1}, - β

be the roots of

a_{1} x^{2} + b_{1} x + c_{1} = 0, a_{1} \neq 0

. Then the quadratic equation whose roots are

α, α_{1}

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If α,β are the roots of ax2+bx+c=0 and −α,γ are the roots of a1x2+b1x+c1=0, then β,γ are roots of Ax2+x+C=0, where C−1=

The correct option is D bc+b1c1Using the formula for sum and product of roots,−C=βγ(γ+β)c1b1=αγ(αγ)−cb=βα(α+β)we can see that 1C=bc+(b1)(c1)

If $α, β$ are the roots of $a x^{2} + b x + c = 0$ and $- α, γ$ are the roots of $a_{1} x^{2} + b_{1} x + c_{1} = 0$ , then $β, γ$ are roots of $A x^{2} + x + C = 0$ , where $C^{- 1} =$

The correct option is D $\frac{b}{c} + \frac{b_{1}}{c_{1}}$
Using the formula for sum and product of roots,
$- C = \frac{β γ}{(γ + β)}$
$\frac{c_{1}}{b_{1}} = \frac{α γ}{(α γ)}$
$\frac{- c}{b} = \frac{β α}{(α + β)}$
we can see that $\frac{1}{C}$ = $\frac{b}{c} + \frac{(b_{1})}{(c_{1})}$