Complex Numbers And Quadratic Equations question 655
Question: If the roots of $ ax^{2}+bx+c=0 $ are $ \sin \alpha $ and $ \cos \alpha $ for some $ \alpha , $ then which one of the following is correct?
Options:
A) $ a^{2}+b^{2}=2ac $
B) $ b^{2}-c^{2}=2ab $
C) $ b^{2}-a^{2}=2ac $
D) $ b^{2}+c^{2}=2ab $
Show Answer
Answer:
Correct Answer: C
Solution:
Let $ \sin \alpha $ and $ \cos \alpha $ be the roots of $ {ax^{2}}+bx+c=0 $ Now, $ \sin \alpha + cos \alpha = \frac{-b}{a} $ and $ \sin \alpha + cos \alpha = \frac{c}{a} $ Consider $ \sin \alpha + cos \alpha = \frac{-b}{a} $ Squaring both side, $ {{( \sin \alpha + cos \alpha )}^{2}} =\frac{b^{2}}{a^{2}} $
$ \Rightarrow {{\sin }^{2}}\alpha + cos^{2} \alpha + 2 sin \alpha cos \alpha = \frac{b^{2}}{a^{2}} $
$ \Rightarrow 1+\frac{2c}{a}=\frac{b^{2}}{a^{2}} $
$ \Rightarrow \frac{a+2c}{a}=\frac{b^{2}}{a^{2}}\Rightarrow a+2c=\frac{b^{2}}{a} $
$ \Rightarrow a^{2}+2ac=b^{2}\Rightarrow b^{2}-a^{2}=2ac $
BETA
