Question

If the points of intersection of the coordinate axes and \(|x+y|=2\) form a rhombus, then its area is

• A. \(8\)
• B. \(16\)
• C. \(2\)
• D. \(4\)

Hint:

For a rhombus formed by coordinate intercepts, first find all intercept points and then use the diagonal formula: \[ \text{Area}=\frac{1}{2}d_1d_2 \]

Answer 1

The Correct Option is A

Step 1: Break the modulus into two lines.
The equation $|x+y|=2$ means either $x+y=2$ or $x+y=-2$. So we are really dealing with two parallel lines.
Step 2: Find where each line meets the axes.
For $x+y=2$: putting $y=0$ gives $(2,0)$, and putting $x=0$ gives $(0,2)$.
Step 3: Find the intercepts of the second line.
For $x+y=-2$: putting $y=0$ gives $(-2,0)$, and putting $x=0$ gives $(0,-2)$.
Step 4: List the four vertices of the figure.
The four points are $(2,0)$, $(0,2)$, $(-2,0)$ and $(0,-2)$. These are symmetric about both axes, forming a rhombus whose diagonals lie along the axes.
Step 5: Measure the diagonals.
One diagonal joins $(2,0)$ and $(-2,0)$, so $d_1 = 4$. The other joins $(0,2)$ and $(0,-2)$, so $d_2 = 4$.
Step 6: Apply the rhombus area formula.
For a rhombus, area $=\frac{1}{2} d_1 d_2 = \frac{1}{2}(4)(4) = 8$ square units.
\[ \boxed{8} \]