Question:
What is the area of the triangle with vertices \(\left(x_1, \dfrac{1}{x_1}\right)\), \(\left(x_2, \dfrac{1}{x_2}\right)\), \(\left(x_3, \dfrac{1}{x_3}\right)\)?
शीर्षों \(\left(x_1, \dfrac{1}{x_1}\right)\), \(\left(x_2, \dfrac{1}{x_2}\right)\), \(\left(x_3, \dfrac{1}{x_3}\right)\) वाले त्रिभुज का क्षेत्रफल क्या है?
Options:
- 0
- \(\left|\dfrac{(x_1-x_2)(x_2-x_3)(x_3-x_1)}{2x_1 x_2 x_3}\right|\) sq units
- \(\left|\dfrac{(x_1-x_2)(x_2-x_3)(x_3-x_1)}{x_1 x_2 x_3}\right|\) sq units
- \(|(x_1-x_2)(x_2-x_3)(x_3-x_1)|\) sq units
- 0
- \(\left|\dfrac{(x_1-x_2)(x_2-x_3)(x_3-x_1)}{2x_1 x_2 x_3}\right|\) वर्ग इकाई
- \(\left|\dfrac{(x_1-x_2)(x_2-x_3)(x_3-x_1)}{x_1 x_2 x_3}\right|\) वर्ग इकाई
- \(|(x_1-x_2)(x_2-x_3)(x_3-x_1)|\) वर्ग इकाई
Correct Answer: b)
Explanation:
Area of triangle \(\Delta = \dfrac{1}{2} \left| \begin{vmatrix} x_1 & 1/x_1 & 1 \\ x_2 & 1/x_2 & 1 \\ x_3 & 1/x_3 & 1 \end{vmatrix} \right|\).
Factoring out \(\dfrac{1}{x_1 x_2 x_3}\) gives the standard Vandermonde determinant result: \(\left| \dfrac{(x_1-x_2)(x_2-x_3)(x_3-x_1)}{2x_1 x_2 x_3} \right|\text{ sq units}\).
त्रिभुज का क्षेत्रफल \(\Delta = \dfrac{1}{2} \left| \begin{vmatrix} x_1 & 1/x_1 & 1 \\ x_2 & 1/x_2 & 1 \\ x_3 & 1/x_3 & 1 \end{vmatrix} \right| = \left| \dfrac{(x_1-x_2)(x_2-x_3)(x_3-x_1)}{2x_1 x_2 x_3} \right|\text{ वर्ग इकाई}\)।