x2−6ax+2−2a+9a2=0, for both the roots to be greater than 3,
Graph is concave upward since
a>0 (+ve)
![](https://search-static.byjusweb.com/question-images/byjus/infinitestudent-images/ckeditor_assets/pictures/38248/content_1.png)
For both roots to exceed 3,
(i)
f(3)>0 (ii)
D≥0 (i)
f(3)>0⇒(3)2−6a(3)+2a−2a+9a2>0⇒9a2−20a+11>0 ⇒9a2−9a−11a+11>0⇒9a(a−1)−11(a−1)>0 ⇒(9a−11)(a−1)>0 ⇒a>1anda>119;aϵ(−∞,1)∪(119,∞) ![](https://search-static.byjusweb.com/question-images/byjus/infinitestudent-images/ckeditor_assets/pictures/38254/content_1.png)
(ii)
D≥0⇒36a2−4(2−2a+9a2)≥0⇒−8+8a≥0⇒a≥1 Combining (i) and (ii);
a>119→ Condition on a for roots to be greater than 3.