Let the area of the triangle with vertices A(1, α), B(α, 0) and C(0, α) be 4 sq. units. If the points (α, - α), (- α, α) and (α², β) are collinear, then β is equal to :

Solution

Maths sir · 3 steps

Use the area to find α², then notice the first two points sit on the line y = -x.

1. Spot the base. A(1, α) and C(0, α) have the same y-coordinate, so AC is horizontal with length 1. The height is the vertical distance from B(α, 0) to the line y = α, which is |α|.

   Area = 1/2 × 1 × |α| = 4 ⇒ |α| = 8 ⇒ α² = 64

2. Line through two points. Both (α, -α) and (-α, α) have y = -x, so they lie on the line y = -x. For (α², β) to be collinear with them, it must also satisfy y = -x.
3. So. β = -α² = -64. This holds for both α = 8 and α = -8, so the answer is (c).

The idea to keep. Before expanding a determinant, look for points with a shared coordinate or an obvious line through them. A pattern like (t, -t) tells you the line is y = -x at once.

Answer from the NTA final answer key. Solution by PrepWiser.