Try this beautiful problem from Algebra PRMO 2017 based on real numbers.

Problem on Real numbers | PRMO | Problem 18


If the real numbers \(x\), \(y\), \(z\) are such that \(x^2 + 4y^2 + 16z^2 = 48\) and \(xy + 4yz + 2zx = 24\). what is the
value of \(x^2 + y^2 + z^2\) ?

  • $24$
  • $21$
  • $34$

Key Concepts


Algebra

Equation

Check the Answer


But try the problem first…

Answer:\(21\)

Source
Suggested Reading

PRMO-2017, Problem 18

Pre College Mathematics

Try with Hints


First hint

The given equation are

\(x^2 + 4y^2 + 16z^2 = 48\)
\(\Rightarrow (x)2 + (2y)2 + (4z)2 = 48\)
\(2xy + 8yz + 4zx = 48\)
adding tis equations we have to solve the problem….

Can you now finish the problem ……….

Second Hint

Now we can say that
\((x)^2 + (2y)^2 + (4z)^2 – (2xy) – (8yz) – (4zx) = 0\)
\(\Rightarrow [(x – 2y)2 + (2y – 4z)2 + (x – 4y)2)] = 0\)
\(x = 2y = 4z \)

\(\Rightarrow \frac{x}{4}=\frac{y}{2}=z\)

Can you finish the problem……..

Final Step

Therefore we may say that,

\((x, y, z) = (4m, 2m, m)\)
\(x^2 + 4y^2 + 16z^2 = 48\)

\(16m^2 + 16m^2 + 16m^2 = 48\)
so \(m^2 = 1\)
\(x^2 + y^2 + z^2 = 21m^2 = 21\)

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