A point P lying inside the curve y=√(2 a x-x^2) is moving...

A point $P$ lying inside the curve $y=\sqrt{2 a x-x^{2}}$ is moving such that its shortest distance from the curve at any position is greater than its distance from $X$ -axis. The point $P$ enclose a region whose area is equal to (a) $\frac{\pi a^{2}}{2}$ (b) $\frac{a^{2}}{3}$ (c) $\frac{2 a^{2}}{3}$ (d) $\left(\frac{3 \pi-4}{6}\right) a^{2}$

A point $P$ lying inside the curve $y=\sqrt{2 a x-x^{2}}$ is moving such that its shortest distance from the curve at any position is greater than its distance from $X$ -axis. The point $P$ enclose a region whose area is equal to
(a) $\frac{\pi a^{2}}{2}$
(b) $\frac{a^{2}}{3}$
(c) $\frac{2 a^{2}}{3}$
(d) $\left(\frac{3 \pi-4}{6}\right) a^{2}$

Key Concepts

Area Calculation in the Plane

Once a region is defined by a set of inequalities, determining its area requires methods from integration or geometric reasoning. This concept ties together the earlier ideas, ultimately leading to the computation of the enclosed area.

Locus and Region Determination

This concept involves finding the set of all points (or the locus) that satisfy given conditions or inequalities. Here, it is used to define a region in the plane based on the relationship between the point's distance from a curve and its distance from the x-axis.

Distance Formula in Coordinate Geometry

This fundamental tool calculates the distance between any two points or between a point and a line in a coordinate plane. It is used to determine both the distance from a point to the circle’s center and the distance from the point to a coordinate axis, which are key comparisons in this problem.

Distance from a Point to a Curve or Circle

This idea deals with computing the smallest (or shortest) distance from a point inside a shape to its boundary. In the case of a circle, it is typically found by subtracting the distance from the point to the circle’s center from the radius of the circle. This concept is central to setting up inequalities that describe a region.

Circle Geometry

This concept involves recognizing that the given curve is part of a circle, understanding its center and radius, and knowing how to express its equation in standard form. It is essential in analyzing properties of the curve and relating points in the plane to the circle.

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