Question

tell me correct answer.. ​

Answer 1

Given:-

A quadratic polynomial is given to us.

The polynomial is ax² +bx + c .

Ratio of its zeroes is m:n .

To Find:-

The option b/w these

b²mn = ( m² + n² ) ac.

( m + n )²ac = b²mn.

b²( m² + n² ) = mnac.

None of these .

Formulae Used:-

Sum of zeroes is given by :

$\large\purple{\underline{\boxed{\pink{\bf{\dag Sum\:of\:zeroes=\dfrac{-(Coefficient\:of\:x)}{Co-efficient\:of\:x^2}}}}}}$

Product of zeroes is given by :

$\large\purple{\underline{\boxed{\pink{\bf{\dag Product\:of\:zeroes=\dfrac{Constant\:term}{(Coefficient\:of\:x^2)}}}}}}$

Answer:-

Given quadratic polynomial to us is ax² + bx + c .

Also given that its roots are in ratio m : n .

Let us take the given ratio be mk : nk .

Now ,

Sum of zeroes = -b/a . ...............(i)

Product of zeroes = c/a . ............(ii)

Let us take the roots be $\alpha$ and $\beta$ .

$\tt{:\implies \alpha + \beta=\dfrac{-b}{a}.}$

$\tt{:\implies mk + nk = \dfrac{-b}{a}.}$

$\tt{:\implies k ( m + n ) =\dfrac{-b}{a}.}$

$\tt{:\implies [k(m+n)]^2=\bigg(\dfrac{-b}{a}\bigg).}$

$\green{\tt{\longmapsto k^2(m+n)^2=\dfrac{b^2}{a^2}..............(iii)}}$

_________________________________________

Also ,

$\tt{:\implies \alpha\beta=\dfrac{c}{a}.}$

$\tt{:\implies mk\times nk =\dfrac{c}{a}.}$

$\green{\tt{\longmapsto k^2mn=\dfrac{c}{a}.............(iv)}}$

$\underline{\pink{\tt{\leadsto Divide\:equ^n\:(iii)\:by\:equ^n\:(iv).}}}$

$\tt{:\implies \dfrac{\cancel{k^2}mn}{\cancel{k^2}(m+n)^2}=\dfrac{c}{\cancel{a}}\times\dfrac{\cancel{a^2}}{b^2}.}$

$\tt{:\implies \dfrac{mn}{(m+n)^2}=\dfrac{ac}{b^2}.}$

$\underline{\boxed{\blue{\tt{\dag b^2mn=(m+n)^2ac}}}}$

•$\underline\red{\pink{\tt{\leadsto Hence\:the\:correct\:option\:is\:(B)\:[b^2mn=(m+n)^2ac].}}}$

Answer 2

Given:-

A quadratic polynomial is given to us.

The polynomial is ax² +bx + c .

Ratio of its zeroes is m:n .

To Find:-

The option b/w these

b²mn = ( m² + n² ) ac.

( m + n )²ac = b²mn.

b²( m² + n² ) = mnac.

None of these .

Formulae Used:-

Sum of zeroes is given by :

\large\purple{\underline{\boxed{\pink{\bf{\dag Sum\:of\:zeroes=\dfrac{-(Coefficient\:of\:x)}{Co-efficient\:of\:x^2}}}}}}

†Sumofzeroes=

Co−efficientofx

2

−(Coefficientofx)

Product of zeroes is given by :

\large\purple{\underline{\boxed{\pink{\bf{\dag Product\:of\:zeroes=\dfrac{Constant\:term}{(Coefficient\:of\:x^2)}}}}}}

†Productofzeroes=

(Coefficientofx

2

)

Constantterm

Answer:-

Given quadratic polynomial to us is ax² + bx + c .

Also given that its roots are in ratio m : n .

Let us take the given ratio be mk : nk .

Now ,

Sum of zeroes = -b/a . ...............(i)

Product of zeroes = c/a . ............(ii)

Let us take the roots be \alphaα and \betaβ .

\tt{:\implies \alpha + \beta=\dfrac{-b}{a}.}:⟹α+β=

a

−b

.

\tt{:\implies mk + nk = \dfrac{-b}{a}.}:⟹mk+nk=

a

−b

.

\tt{:\implies k ( m + n ) =\dfrac{-b}{a}.}:⟹k(m+n)=

a

−b

.

\tt{:\implies [k(m+n)]^2=\bigg(\dfrac{-b}{a}\bigg).}:⟹[k(m+n)]

2

=(

a

−b

).

\green{\tt{\longmapsto k^2(m+n)^2=\dfrac{b^2}{a^2}..............(iii)}}⟼k

2

(m+n)

2

=

a

2

b

2

..............(iii)

_________________________________________

Also ,

\tt{:\implies \alpha\beta=\dfrac{c}{a}.}:⟹αβ=

a

c

.

\tt{:\implies mk\times nk =\dfrac{c}{a}.}:⟹mk×nk=

a

c

.

\green{\tt{\longmapsto k^2mn=\dfrac{c}{a}.............(iv)}}⟼k

2

mn=

a

c

.............(iv)

\underline{\pink{\tt{\leadsto Divide\:equ^n\:(iii)\:by\:equ^n\:(iv).}}}

⇝Divideequ

n

(iii)byequ

n

(iv).

\tt{:\implies \dfrac{\cancel{k^2}mn}{\cancel{k^2}(m+n)^2}=\dfrac{c}{\cancel{a}}\times\dfrac{\cancel{a^2}}{b^2}.}:⟹

k

2

(m+n)

2

k

2

mn

=

a

c

× b 2a 2

\tt{:\implies \dfrac{mn}{(m+n)^2}=\dfrac{ac}{b^2}.}:⟹

(m+n)

2

mn

= b 2ac

.

\underline{\boxed{\blue{\tt{\dag b^2mn=(m+n)^2ac}}}}

†b 2

mn=(m+n)

2 ac

•\underline\red{\pink{\tt{\leadsto Hence\:the\:correct\:option\:is\:(B)\:[b^2mn=(m+n)^2ac].}}}

⇝Hencethecorrectoptionis(B)[b

2

mn=(m+n)

2 ac].