Question

Let A \((x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let B \((1, 4, -1)\) and C \((2, 0, -2)\). Then among the statements:
(S1): ABC is an isosceles right angled triangle, and
(S2): the area of \(\triangle ABC\) is \( \frac{9\sqrt{2}}{2} \).

• A. only (S1) is true
• B. both are true
• C. only (S2) is true
• D. both are false

Hint:

Understanding the geometric properties and applying distance and area formulas accurately are crucial in solving these types of problems.

Answer 1

The Correct Option is D

Step 1: Calculate the distances. Calculate distances between \(A\), \(B\), and \(C\) to verify if \(ABC\) forms an isosceles right triangle. 

Step 2: Verify statement (S1). Use distance formulas to find \(AB\), \(BC\), and \(CA\) and check for equality and Pythagorean theorem. 

Step 3: Verify statement (S2). Calculate the area of \(\triangle ABC\) using the determinant method or Heron's formula to see if it matches \( \frac{9\sqrt{2}}{2} \). 

Step 4: Conclusion for each statement. Determine the truth of each statement based on calculations. 

Conclusion: After performing the calculations, both statements are found to be false.