Question:
Find the value of \(x^2 + y^2 + z^2\) if \(x + y + z = 15\) and \(xy + yz + zx = 54\).
यदि \(x + y + z = 15\) और \(xy + yz + zx = 54\) है, तो \(x^2 + y^2 + z^2\) का मान ज्ञात कीजिए।
Options:
- 117
- 99
- 81
- 135
- 117
- 99
- 81
- 135
Correct Answer: a)
Explanation:
Using the identity:
\((x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)\)
\(15^2 = x^2 + y^2 + z^2 + 2(54)\)
\(225 = x^2 + y^2 + z^2 + 108\)
\(x^2 + y^2 + z^2 = 225 – 108 = 117\).
\((x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)\)
\(15^2 = x^2 + y^2 + z^2 + 2(54)\)
\(225 = x^2 + y^2 + z^2 + 108 \implies x^2 + y^2 + z^2 = 117\)।