It is given that
(x−1)2+2kx+11=0 has no real roots. (Where
k is the largest integer)
(x−1)2+2kx+11=0, which can be written as:
‌⇒x2−2x+1+2kx+11=0‌⇒x2−2(k−1)x+12=0We know that for no real roots,
D<0⇒b∧2−4ac<0Hence,
{2(k−1)}2−4⋅1⋅12<0‌⇒4(k−1)2<48‌⇒(k−1)2<12Since
k is an integer, it implies (
k−1 ) is also an integer.
Therefore, from the above inequality, we can say that the largest possible value of
(k−1)=3⇒ The largest possible value of
k is 4 .
Now we need to calculate the least possible value of
‌+9y.
‌+9y can be written as
+9y=‌+9y The least possible value of
9y+‌ can be calculated using A.M-G.M inequality.
Using A.M-G.M inequality, we get:
‌‌≥√9y×‌⇒‌≥√9⇒‌≥3⇒9y+‌≥6Hence, the least possible value is 6