Question

The co-ordinates of the vertices of a triangle are [m(m + 1), (m + 1)], [(m + 1)(m + 2), (m + 2)] and [(m + 2)(m + 3), (m + 3)]. Then which one among the following is correct?

(a) The area of the triangle is dependent on m

(b) The area of the triangle is independent on m

(c) Answer cannot be predicted

(d) Data inadequate

The question was asked by my school principal while I was bunking the class.

This interesting question is from Application of Determinants in chapter Determinants of Mathematics – Class 12

Answer 1

The correct answer is (b) The area of the triangle is independent on m

Explanation: The area o the triangle with the given point as vertices is,

1/2 \(\begin{vmatrix}m(m+1) & (m+1) & 1 \\(m+1)(m+2) & (m+2) & 1 \\(m+2)(m+3) & (m+3) & 1 \end {vmatrix}\)

 = 1/2 \(\begin{vmatrix}m^2 + m & (m+1) & 1 \\m^2 + 3m + 2 & (m+2) & 1 \\m^2 + 5m + 6 & (m+3) & 1 \end {vmatrix}\)

Now, by performing the row operation R2 = R2 – R1 and R3 = R3 – R2

= 1/2 \(\begin{vmatrix}m^2 + m & (m+1) & 1 \\2m + 2 & 1 & 0 \\2m + 4 & 1 & 0 \end {vmatrix}\)

Now, breaking the determinant we get,

= 1/2 (2m + 2 – 2m – 4)

= -1

Thus, it is independent of m.