Question:hard

The area, in sq. units, enclosed by the lines \(x=2\), \(y=|x-2|+4\), the \(X\)-axis and the \(Y\)-axis is equal to

Updated On: Jan 15, 2026
  • 8
  • 12
  • 10
  • 6
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The Correct Option is C

Solution and Explanation

To determine the area bounded by the lines \(x=2\), \(y=|x-2|+4\), the \(X\)-axis, and the \(Y\)-axis, we analyze the geometry. The equation \(y=|x-2|+4\) defines a V-shaped graph, which can be expressed as two linear equations:

  • For \(x \geq 2\), \(y=x-2+4\), simplifying to \(y=x+2\).
  • For \(x < 2\), \(y=-(x-2)+4\), simplifying to \(y=-x+6\).

The two lines are \(y=x+2\) and \(y=-x+6\). Their intersection with \(x=2\) is calculated as follows:

  • With \(y=x+2\), at \(x=2\), \(y=2+2=4\).
  • With \(y=-x+6\), at \(x=2\), \(y=-2+6=4\).

Both lines intersect \(x=2\) at \(y=4\), indicating the vertex of the V is on this line. The line \(y=x+2\) intersects the \(X\)-axis at \(x=-2\) (when \(y=0\)). The line \(y=-x+6\) intersects the \(Y\)-axis at \(y=6\) (when \(x=0\)).

The area of the triangle formed by vertices \((0,0)\), \((0,6)\), and \((-2,0)\) is calculated. The base along the \(Y\)-axis from \((0,0)\) to \((0,6)\) is 6 units. The height along the \(X\)-axis from \((0,0)\) to \((-2,0)\) is 2 units.

The area of a triangle is given by:

\[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}=\frac{1}{2} \times 6 \times 2=6\text{ units}^2\]

Additionally, the region includes another triangle with vertices: \((0,0)\), \((0,6)\), and \((2,4)\).

Vertex 1Vertex 2Vertex 3
\((0,0)\)\((0,6)\)\((2,4)\)

The area of this triangle is calculated. The base from \((0,6)\) to \((0,0)\) is 6 units. The height from \((2,4)\) to the \(Y\)-axis is 2 units.

Area = \(\frac{1}{2} \times\ 6 \times\ 2 = 6\text{ units}^2\).

The total enclosed area is the sum of these two areas: \(6 + 4 = 10 \text{ sq. units}\).

Therefore, the area enclosed by the specified lines is \(\boxed{10}\text{ sq. units.}‍\)

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