A) 18, 50/9
B) 18, 25/9
C) 27, 50/9
D) None of these
Correct Answer: A
Solution :
For real roots\[D\ge 0\] \[{{(k-2)}^{2}}-4({{k}^{2}}+3k+5)\ge 0\] \[({{k}^{2}}{{+}^{2}}-4k)-4{{k}^{2}}-12k-20\ge 0\] \[-3{{k}^{2}}-16k-16\ge 0;3{{k}^{2}}+16k+16\le 0\]\[\left( k+\frac{4}{3} \right)(k+4)\le 0\] Now \[E={{\alpha }^{2}}+{{\beta }^{2}};\] \[E={{(\alpha +\beta )}^{2}}-2\alpha \beta \] \[E={{(k-2)}^{2}}-2({{k}^{2}}+3+5)=-{{k}^{2}}-10k-6\] \[E=-({{k}^{2}}-10k+6)=-[{{(k+5)}^{2}}-19]=19-{{(k+5)}^{2}}\]\[\therefore \]\[{{E}_{\min }}=19-\frac{121}{9}=\frac{171-121}{9}=\frac{50}{9}\] \[{{E}_{\max }}\]occurs when k =- 4 \[{{E}_{\max }}\]\[=19-1=18\]You need to login to perform this action.
You will be redirected in
3 sec