Question

\(\left|\begin{matrix}x_1 & y_1\\ x_2 & y_2\end{matrix}\right|\) represents the area of

• A. Circle
• B. Ellipse
• C. Triangle
• D. Parallelogram

Hint:

A \(2\times2\) determinant gives the area of a parallelogram. Half of it gives the area of a triangle.

Answer 1

The Correct Option is D

Concept:
A determinant of order \(2\) formed by two vectors represents the signed area of the parallelogram formed by those two vectors. If the vectors are: \[ \vec{a}=x_1\hat{i}+y_1\hat{j} \] and \[ \vec{b}=x_2\hat{i}+y_2\hat{j} \] then the area of the parallelogram formed by these two vectors is: \[ \left|\begin{matrix}x_1 & y_1 \\ x_2 & y_2\end{matrix}\right| \]

Step 1: Write the determinant.
The given determinant is: \[ \left|\begin{matrix}x_1 & y_1 \\ x_2 & y_2\end{matrix}\right| \] Expanding it: \[ =x_1y_2-x_2y_1 \] The absolute value of this expression gives the area.

Step 2: Connect determinant with geometry.
The determinant: \[ x_1y_2-x_2y_1 \] represents the cross product magnitude of two vectors in two dimensions. The magnitude of cross product gives the area of the parallelogram formed by the two vectors.

Step 3: Difference between triangle and parallelogram area.
If the same two vectors form a triangle, then the area of the triangle is half of the parallelogram area: \[ \text{Area of triangle}=\frac12\left|\begin{matrix}x_1 & y_1 \\ x_2 & y_2\end{matrix}\right| \] But the determinant itself represents the parallelogram area.

Step 4: Check the options.
Option (A) Circle is incorrect because circle area does not use this determinant.
Option (B) Ellipse is incorrect.
Option (C) Triangle is incorrect because triangle area is half of this determinant.
Option (D) Parallelogram is correct. Hence, the correct answer is: \[ \boxed{(D)\ \text{Parallelogram}} \]