Question

Let $\alpha, \beta, \gamma$ be distinct real numbers such that
$a \alpha^2+ b \alpha+ c =(\sin \theta) \alpha^2+(\cos \theta) \alpha $
$a \beta^2+ b \beta+ c =(\sin \theta) \beta^2+(\cos \theta) \beta$
$a \gamma^2+ b \gamma+ c =(\sin \theta) \gamma^2+(\cos \theta) \gamma$
(where a, b, c, $\in$ R.)
The maximum value of the expression $\frac{a^2+b^2}{a^2+3 a b+5 b^2}$ is equal to

• A. 1
• B. 2
• C. $\frac{5}{2}$
• D. $\frac{7}{2}$

Answer 1

Answer: (B)

The 3 given equation are suggestive that $\alpha, \beta$ and $\gamma$ are the roots of
$ax ^2+ bx + c =(\sin \theta) x ^2+(\cos \theta) x \text { or }( a -\sin \theta) x ^2+( b -\cos \theta) x + c =0 $....(1)
$\therefore a =\sin \theta, b =\cos \theta \text { and } c =0 \text { (Using concept of identity) } $
$E=\frac{a^2+b^2}{a^2+3 a b+5 b^2}=\frac{1}{\sin ^2 \theta+3 \sin \theta \cos \theta+5 \cos ^2 \theta} $
$=\frac{2}{1-\cos 2 \theta+3 \sin 2 \theta+5(1+\cos 2 \theta)}=\frac{2}{6+3 \sin 2 \theta+4 \cos 2 \theta}$
$\therefore E _{\max }=\frac{2}{6-5}=2$