Question

If \( S \) is the set of all real values of \( a \) such that a plane passing through the points \( (-a^2, 1, 1), (1, -a^2, 1), (1, 1, -a^2) \) also passes through the point \( (-1, -1, 1) \), then \( S = \dots \)

• A. \( \{\sqrt{3}\} \)
• B. \( \{\sqrt{3}, -\sqrt{3}\} \)
• C. \( \{1, -1\} \)
• D. \( \{3, -3\} \)

Hint:

To find the equation of a plane passing through three points, substitute the coordinates into the general equation of a plane and solve for the coefficients.

Answer 1

The Correct Option is B

We are given three points: \( (-a^2, 1, 1) \), \( (1, -a^2, 1) \), and \( (1, 1, -a^2) \), and the condition that the plane passing through these points also passes through the point \( (-1, -1, 1) \). Step 1: To find the equation of the plane passing through the three given points, we can use the general form of the plane equation: \[ Ax + By + Cz + D = 0 \] Substitute the coordinates of the given points into the equation to find the values of \( A \), \( B \), \( C \), and \( D \). Step 2: Substitute the coordinates of the point \( (-1, -1, 1) \) into the equation of the plane to find the condition for \( a \). Step 3: Solving this equation, we find that the values of \( a \) that satisfy the condition are \( \pm \sqrt{3} \). Thus, the set \( S \) is: \[ S = \{\sqrt{3}, -\sqrt{3}\} \] % Final Answer The set \( S = \{\sqrt{3}, -\sqrt{3}\} \).