GATE 2021 BT – Question 51
A triangle with vertices $(k,0),(2,0),(0,-2)$ has area 2. The value of k is _______.
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Correct answer: 0 to 0 OR 4 to 4
Explanation
The official key accepts either 0 to 0 OR 4 to 4. These are separate accepted answers, not one continuous interval.
The area of a triangle with vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ is
$$A=\tfrac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right| .$$
With $(k,0),(2,0),(0,-2)$:
$$A=\tfrac12\left|k(0+2)+2(-2-0)+0\right|=\tfrac12|2k-4|=|k-2| .$$
(Geometrically, the base along the x-axis is $|k-2|$ and the height is 2, so $A=\tfrac12\times|k-2|\times2$.)
**Set the area to 2:** $|k-2|=2$, so $k=4$ or $k=0$.
The official answer key gives **k = 4**. (The value $k=0$ also gives a triangle of area 2, so the problem as stated has two solutions, and 4 is the one in the key.)
Official GATE 2021 answer key: https://gate2026.iitg.ac.in/doc/download/Answer_keys2021/bt_2021.pdf