Question

Solve this :::::::;;;;;;;;;!!!!;!!!;!;;!!;;;​

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

1. b²mn = ( m² + n² ) ac.
2. ( m + n )²ac = b²mn.
3. b²( m² + n² ) = mnac.
4. 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 $\beta$ .

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

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

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

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

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

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Also ,

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

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

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

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

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

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

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

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