If - - be the roots of x2-p x-q-0 and -h- -h are the roots of x2-r x-s-0- thenA- p-r-q-s2 h-p-q-r-s-C- p2-4 q-r2-4 sD- p r2-q s2

The correct option is C α + β= p, αβ=q α+β+2h= r, (α + h)(β+h)=s p+2h= r ⇒ h=p r/2 ........(i) Now, αβ +h(α + β)+h^2=s ⇒ q+h( p)+h^2=s ⇒ q+ ( p ...

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Standard XII

Mathematics

Equality of 2 Complex Numbers

If α, β be th...

Question

If $α, β$ be the roots of $x^{2}$ +px+q=0 and $α + h, β + h$ are the roots of $x^{2}$ +rx+s=0 , then

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Solution

The correct option is C

$α + β = - p, α β = q α + β + 2 h = - r, (α + h) (β + h) = s$

$- p + 2 h = - r \Rightarrow h = \frac{p - r}{2}$ ........(i)

$N o w, α β + h (α + β) + h^{2} = s \Rightarrow q + h (- p) + h^{2} = s \Rightarrow q + (\frac{p - r}{2}) (- p) + {(\frac{p - r}{2})}^{2} = s \Rightarrow q + \frac{(p^{2} - p r)}{2} + \frac{p^{2} + r^{2} - 2 p r}{4} = s$

$\Rightarrow 4 q - 2 p^{2} + 2 p r + p^{2} + r^{2} - 2 p r = 4 s \Rightarrow 4 q - p^{2} + r^{2} - 4 s = 0 \Rightarrow r^{2} - 4 s = p^{2} - 4 q$

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If α,β be the roots of x2+px+q=0 and α+h, β+h are the roots of x2+rx+s=0 , then

The correct option is C α+β=−p, αβ=qα+β+2h=−r,(α+h)(β+h)=s −p+2h=−r⇒h=p−r2 ........(i) Now, αβ+h(α+β)+h2=s⇒q+h(−p)+h2=s⇒q+(p−r2)(−p)+(p−r2)2=s⇒q+(p2−pr)2+p2+r2−2pr4=s ⇒4q−2p2+2pr+p2+r2−2pr=4s⇒4q−p2+r2−4s=0⇒r2−4s=p2−4q

If $α, β$ be the roots of $x^{2}$ +px+q=0 and $α + h, β + h$ are the roots of $x^{2}$ +rx+s=0 , then