Question:
If the points of intersection of the coordinate axes and \(|x+y|=2\) form a rhombus, then its area is
If the points of intersection of the coordinate axes and \(|x+y|=2\) form a rhombus, then its area is
Show Hint
For a rhombus formed by coordinate intercepts, first find all intercept points and then use the diagonal formula:
\[
\text{Area}=\frac{1}{2}d_1d_2
\]
Updated On: Jun 22, 2026
- \(8\)
- \(16\)
- \(2\)
- \(4\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Rewrite the given equation.
The equation
\[ |x+y|=2 \] can be written as two straight lines:
\[ x+y=2 \] and
\[ x+y=-2 \]
Step 2: Find the intercepts on the coordinate axes.
For the line
\[ x+y=2 \] When \(y=0\),
\[ x=2 \] So, one intercept is \((2,0)\).
When \(x=0\),
\[ y=2 \] So, another intercept is \((0,2)\).
For the line
\[ x+y=-2 \] When \(y=0\),
\[ x=-2 \] So, one intercept is \((-2,0)\).
When \(x=0\),
\[ y=-2 \] So, another intercept is \((0,-2)\).
Thus, the four vertices of the rhombus are
\[ (2,0),\;(0,2),\;(-2,0),\;(0,-2) \]
Step 3: Find the diagonals of the rhombus.
The horizontal diagonal is from \((-2,0)\) to \((2,0)\).
Its length is
\[ 4 \]
The vertical diagonal is from \((0,-2)\) to \((0,2)\).
Its length is
\[ 4 \]
Step 4: Use the area formula of a rhombus.
Area of a rhombus is given by
\[ \text{Area}=\frac{1}{2}\times d_1 \times d_2 \] where \(d_1\) and \(d_2\) are the diagonals.
Substituting the values,
\[ \text{Area}=\frac{1}{2}\times 4 \times 4 \] \[ =8 \]
Step 5: Final conclusion.
Hence, the area of the rhombus is
\[ \boxed{8} \]
The equation
\[ |x+y|=2 \] can be written as two straight lines:
\[ x+y=2 \] and
\[ x+y=-2 \]
Step 2: Find the intercepts on the coordinate axes.
For the line
\[ x+y=2 \] When \(y=0\),
\[ x=2 \] So, one intercept is \((2,0)\).
When \(x=0\),
\[ y=2 \] So, another intercept is \((0,2)\).
For the line
\[ x+y=-2 \] When \(y=0\),
\[ x=-2 \] So, one intercept is \((-2,0)\).
When \(x=0\),
\[ y=-2 \] So, another intercept is \((0,-2)\).
Thus, the four vertices of the rhombus are
\[ (2,0),\;(0,2),\;(-2,0),\;(0,-2) \]
Step 3: Find the diagonals of the rhombus.
The horizontal diagonal is from \((-2,0)\) to \((2,0)\).
Its length is
\[ 4 \]
The vertical diagonal is from \((0,-2)\) to \((0,2)\).
Its length is
\[ 4 \]
Step 4: Use the area formula of a rhombus.
Area of a rhombus is given by
\[ \text{Area}=\frac{1}{2}\times d_1 \times d_2 \] where \(d_1\) and \(d_2\) are the diagonals.
Substituting the values,
\[ \text{Area}=\frac{1}{2}\times 4 \times 4 \] \[ =8 \]
Step 5: Final conclusion.
Hence, the area of the rhombus is
\[ \boxed{8} \]
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