Question:
Let O be the origin, and P and Q be two points on the rectangular hyperbola \( xy = 12 \) such that the midpoint of the line segment PQ is \( \left( \frac{1}{2}, -\frac{1}{2} \right) \). Then the area of the triangle OPQ equals:
Let O be the origin, and P and Q be two points on the rectangular hyperbola \( xy = 12 \) such that the midpoint of the line segment PQ is \( \left( \frac{1}{2}, -\frac{1}{2} \right) \). Then the area of the triangle OPQ equals:
Updated On: Apr 10, 2026
- \( \frac{3}{2} \)
- \( \frac{5}{2} \)
- \( \frac{7}{2} \)
- \( \frac{9}{2} \)
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The Correct Option is B
Solution and Explanation
Step 1: Equation of the hyperbola.
The equation of the rectangular hyperbola is given as \( xy = 12 \). The midpoint of the points P and Q is given as \( \left( \frac{1}{2}, -\frac{1}{2} \right) \). Using the midpoint formula, we calculate the coordinates of points P and Q.
Step 2: Calculate the area of the triangle.
The area of triangle OPQ is given by the formula: \[ \text{Area} = \frac{1}{2} \times \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \] Substitute the coordinates of O, P, and Q into this formula to calculate the area of triangle OPQ.
Final Answer: \( \frac{5}{2} \)
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