First slide

Theory of equations

Question

If $α, β$ are the roots of $x^{2} - px + q = 0$ and $α^{'}, β^{'}$ are the roots of $x^{2} - p^{'} x + q^{'} = 0$ then the value of ${(α - α^{'})}^{2} + {(β - α^{'})}^{2} + {(α - β^{'})}^{2} + {(β - β^{'})}^{2}$ is

Moderate

A

$2 \{p^{2} - 2 q + p^{' 2} - 2 q^{'} - {pp}^{'}\}$
B

$2 \{p^{2} - 2 q + p^{' 2} - 2 q^{'} + {qq}^{'}\}$
C

$2 \{p^{2} - 2 q - p^{' 2} - 2 q^{'} + {pp}^{'}\}$
D

$2 \{p^{2} - 2 q - p^{' 2} - 2 q^{'} - {qq}^{'}\}$

Solution

Since $α, β$ are the roots of $x^{2} - px + q = 0$
and $α^{'}, β^{'}$ are the roots of $x^{2} - p^{'} x + q^{'} = 0$
$α + β = p, αβ = q, α^{'} + β^{'} = p^{'}, α^{'} β^{'} = q^{'}$
Now, ${(α - α^{'})}^{2} + {(β - α^{'})}^{2} + {(α - β^{'})}^{2} + {(β - β^{'})}^{2}$
$\begin{array}{l} = 2 (α^{2} + β^{2}) + 2 (α^{' 2} + β^{' 2}) - 2 α^{'} (α + β) - 2 β^{'} (α + β) \\ = 2 \{(α + β)^{2} - 2 αβ + {(α^{'} + β^{'})}^{2} - 2 α^{'} β^{'} - (α + β) (α^{'} + β^{'})\} \\ = 2 (p^{2} - 2 q + p^{' 2} - 2 q^{'} - {pp}^{'}) \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App