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Graph of Quadratic Expression

lf α, β are...

Question

lf $α, β$ are the roots of $x^{2} - p x - c - p = 0$ , then $\frac{α^{2} + 2 α + 1}{α^{2} + 2 α + c} + \frac{β^{2} + 2 β + 1}{β^{2} + 2 β + c} =$

A

3

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B

2

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C

1

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D

0

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Solution

The correct option is A $1$
The given equation can be written as,
$x^{2} - p x - (p + c) = 0$
The basic properties of quadratic equations says,
$α + β = p$ and $α β = - (p + c)$
From above two equation we get,
$c = - α - β + α β$
Now according to the question,
$\frac{(α^{2} + 2 α + 1)}{(α^{2} - 2 α + c)} + \frac{(β^{2} + 2 β + 1)}{(β^{2} + 2 β + c)}$
Then expression gets reduced to:
$\frac{(α^{2} + 2 α + 1)}{(α^{2} - 2 α - α - β - α β)} + \frac{(β^{2} + 2 β + 1)}{(β^{2} + 2 β - β - α - α β)}$

$= \frac{(α + 1)^{2}}{(α + 1) (α - β)} - \frac{(β + 1)^{2}}{(β + 1) (α - β)}$

$= \frac{(α + 1)}{(α - β)} - \frac{(β + 1)}{(α - β)}$

$= \frac{(α + 1 - β - 1)}{(α - β)}$

$= \frac{(α - β)}{(α - β)}$

$= 1$

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