The ex-radii of triangle are 10 , 12 and 14. If the sides of the triangle are the roots of the cubic x3 – px2 + qx – = 0, where p, q, are integers, find the integer nearest to p+q +r .
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. r1 = = , r2 = = 12.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. S–a 2 S–b.
Hint 3: Proceed with the final algebraic steps to solve the system. and r3 = solve for the final value.
Step 1: r1 = = , r2 = = 12
Step 2: S–a 2 S–b
Step 3: and r3 = = 14
Step 4: On solving above equations, we get
Step 5: a = 13, = 14, = 15
Step 6: Let f(x) = x3 – px2 + qx – r
Step 7: = (x – 13) (x – 14) (x – 15)
Step 8: f(–1) = –1 – – – r
Step 9: = (–14) (–15) (–16)
Step 10: => + + = 14 15 16 – 1
Step 11: => + + = 3364 58
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