Determine the number of positive integral value of p for which there exists a triangle with side a, b and c which satisfy a^2+ (p^2+9 ) b^2+9 c^2-6 a b-6 p b c=0

Step-by-step solution

aligned & a^2+9 b^2-6 a b+p^2 b^2-6 p b c+9 c^2=0 & (a-3 b)^2+(p b-3 c)^2=0 & a=3 b p b=3 c & c= p b 3 aligned If a is the largest side aligned & b+ p b 3 >3 b & 3+p>9 aligned p>6 ......(i) If c is largest side 3 b+b> p b 3 p < 12 ......(ii) so, p= 7,8,9,10,11 Number of positive integral value of p is 5