If the area bounded by the corve y=x^2+1, y=x and the pair...

If the area bounded by the corve $y=x^{2}+1, y=x$ and the pair of lines $x^{2}+y^{2}+2 x y-4 x-4 y+3=0$ is $K$ units, then the area of the region bounded by the curve $y=x^{2}+1, y=\sqrt{x-1}$ and the pair of lines $(x+y-1)(x+y-3)=0$, is (a) $K$ (b) $2 \mathrm{~K}$ (c) $\frac{K}{2}$ (d) None of these

If the area bounded by the corve $y=x^{2}+1, y=x$ and the pair of lines $x^{2}+y^{2}+2 x y-4 x-4 y+3=0$ is $K$ units, then the area of the region bounded by the curve $y=x^{2}+1, y=\sqrt{x-1}$ and the pair of lines $(x+y-1)(x+y-3)=0$, is
(a) $K$
(b) $2 \mathrm{~K}$
(c) $\frac{K}{2}$
(d) None of these

Key Concepts

Understanding Conic Sections and Degenerate Cases

In problems where equations of conic sections are involved, it is important to recognize when these equations represent degenerate cases such as pairs of straight lines. Recognizing this allows one to simplify the problem by treating parts of the boundary as simple linear equations rather than more complex curves.

Coordinate Transformations and Geometric Similarity

This concept deals with understanding how transformations, such as translations and rotations, affect the regions defined by given equations. Recognizing geometric similarity or symmetry between different regions can lead to relationships between their areas, such as scaling by a constant factor. This insight is useful in comparing areas under different but related sets of curves.

Area Calculation Using Integration

This concept involves computing the area of a region bounded by curves through integration. It requires setting up definite integrals by first determining the limits from the intersection points of the curves, and then integrating the difference of the functions over the appropriate interval. This technique is central to solving problems in coordinate geometry where regions are defined by complex boundaries.

Finding Intersection Points

Determining where curves intersect is essential for identifying the limits of integration and understanding the shape of the bounded region. By solving the equations that represent the curves simultaneously, one can find the exact points of intersection, which serve as the boundaries of the area to be calculated.

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