If $α, β$ are roots of the equation $x^{2} - 15 x + 1 = 0$ . Then the value of ${(\frac{1}{α} - 15)}^{- 2} + {(\frac{1}{β} - 15)}^{- 2}$ is

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225

223

227

900

answer is C.

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Detailed Solution

The given equation $x^{2} - 15 x + 1 = 0$

$α, β$ are the roots given equation

So $α + β = 15, α β = 1$

if $x = α$ , Then $x^{2} - 15 x + 1 = 0$ will be

$α^{2} - 15 α + 1 = 0 \Rightarrow \frac{1}{α} - 15 = - α$

Squaring on both sides

$\Rightarrow {(\frac{1}{α} - 15)}^{2} = α^{2}$ …… ( 1 )

if $x = β$ , Then $x^{2} - 15 x + 1 = 0$ will be

$β^{2} - 15 β + 1 = 0 \Rightarrow \frac{1}{β} - 15 = - β$

Squaring on both sides

$\Rightarrow {(\frac{1}{β} - 15)}^{2} = β^{2}$ …… ( 2 )

adding eq 1 and 2

Now ${(\frac{1}{α} - 15)}^{- 2} + {(\frac{1}{α} - 15)}^{- 2}$ = $\frac{1}{α^{2}} + \frac{1}{β^{2}}$

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

= $\frac{225 - 2}{1} = 223$ ( $∵$ $α + β = 15, α β = 1$ )

$∴$ ${(\frac{1}{α} - 15)}^{- 2} + {(\frac{1}{β} - 15)}^{- 2}$ = 223

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