Determine the number of positive integral value of for which there exists triangle with side a, and which satisfy a2 + (p2 + 9)b2 + 9c2 – 6ab – 6pbc = 0
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. a2 + 9b2 – 6ab + p2b2 – 6pbc + 9c2 = 0.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. (a – 3b)2 + (pb – 3c)2 = 0.
Hint 3: Proceed with the final algebraic steps to solve the system. a solve for the final valueb pb solve for the final valuec.
Step 1: a2 + 9b2 – 6ab + p2b2 – 6pbc + 9c2 = 0
Step 2: (a – 3b)2 + (pb – 3c)2 = 0
Step 3: a = 3b pb = 3c
Step 4: If is the largest side
Step 5: b+ 3b
Step 6: p>6 …(i)
Step 7: If is largest side
Step 8: p < 12 …(ii)
Step 9: so, = {7, 8, 9, 10, 11}
Step 10: => Number of positive integral value of is 5
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