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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Determine the number of positive integral value of p for which...

Determine the number of positive integral value of pp for which there exists a triangle with side a,ba, b and cc which satisfy a2+(p2+9)b2+9c2−6ab−6pbc=0a^2+\left(p^2+9\right) b^2+9 c^2-6 a b-6 p b c=0

Correct answer

5

Step-by-step solution

a2+9b2−6ab+p2b2−6pbc+9c2=0(a−3b)2+(pb−3c)2=0a=3bpb=3c⇒c=pb3\begin{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 \quad p b=3 c \\ & \Rightarrow \quad c=\frac{p b}{3} \end{aligned} If aa is the largest side b+pb3>3b3+p>9\begin{aligned} & b+\frac{p b}{3}>3 b \\ & 3+p>9 \end{aligned} p>6p>6......(i)IfcIf c is largest side 3b+b>pb33 b+b>\frac{p b}{3} p<12p < 12......(ii) so, p={7,8,9,10,11}p=\{7,8,9,10,11\} ⇒\Rightarrow \quad Number of positive integral value of pp is 5
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