Question

If ${}^{\prime }{\alpha }^{\prime }{,}^{\prime }{\beta }^{\prime }$ are the roots of $x^2-15x+1=0$ then, the value of ${\left(\frac{1}{\alpha }-15\right)}^{-2}+{\left(\frac{1}{\beta }-15\right)}^{-2}$

• A. $225$
• B. $223$
• C. $900$
• D. $850$

Answer 1

Answer: (B)

$223$

$\alpha ,\beta$ are the roots of $x^2-15x+1=0⟹{\alpha }^2-15\alpha +1=0⟹\frac{1}{\alpha }-15=-\alpha$

$\alpha +\beta =15,\alpha \beta =1$

$(\frac{1}{\alpha }-15{)}^{-2}+(\frac{1}{\beta }-15{)}^{-2}={\alpha }^{-2}+{\beta }^{-2}=\frac{{\alpha }^2+{\beta }^2}{{\alpha }^2{\beta }^2}=\frac{{15}^2-2}{1}=225-2=223$