A man is walking along the boundary of the circular path,...

A man is walking along the boundary of the circular path, $x^{2}+y^{2}-2 x-2 y=0$. If the speed of the man is $\frac{\pi}{3} \mathrm{~m} / \mathrm{sec}$ and he starts from $\mathrm{P}(2,2)$ and moves in the anti-clockwise direction, then the position of the man after $\frac{5}{\sqrt{2}}$ sec is (a) $\left(\frac{1-\sqrt{3}}{2}, \frac{3-\sqrt{3}}{2}\right)$ (b) $\left(\frac{3-\sqrt{3}}{2}, \frac{3+\sqrt{3}}{2}\right)$ (c) $\left(\frac{-3+\sqrt{3}}{2}, \frac{-3-\sqrt{3}}{2}\right)$ (d) $\left(\frac{-1-\sqrt{3}}{2}, \frac{-3+\sqrt{3}}{2}\right)$

A man is walking along the boundary of the circular path, $x^{2}+y^{2}-2 x-2 y=0$. If the speed of the man is $\frac{\pi}{3} \mathrm{~m} / \mathrm{sec}$
and he starts from $\mathrm{P}(2,2)$ and moves in the anti-clockwise direction, then the position of the man after $\frac{5}{\sqrt{2}}$ sec is
(a) $\left(\frac{1-\sqrt{3}}{2}, \frac{3-\sqrt{3}}{2}\right)$
(b) $\left(\frac{3-\sqrt{3}}{2}, \frac{3+\sqrt{3}}{2}\right)$
(c) $\left(\frac{-3+\sqrt{3}}{2}, \frac{-3-\sqrt{3}}{2}\right)$
(d) $\left(\frac{-1-\sqrt{3}}{2}, \frac{-3+\sqrt{3}}{2}\right)$

Key Concepts

Relationship between Linear and Angular Speed

In circular motion, linear speed (the distance traveled per unit time) and angular speed (the change in the angular parameter per unit time) are related by the formula v = r?, where v is linear speed, r is the radius of the circle, and ? is the angular speed in radians per unit time. This relationship allows for the conversion of a given linear speed to the corresponding angular displacement over a period of time, which is essential for analyzing motion on a circular path.

Parametric Representation of Circular Motion

A circle can be described parametrically using trigonometric functions: x = h + r cos(?) and y = k + r sin(?), where (h, k) is the center, r is the radius, and ? is the parameter (angle). This representation is particularly useful for dealing with problems of motion along a circle, as it directly relates the position on the circle to the angle, making it easier to compute the location after a given angular displacement.

Standard Equation of a Circle

The standard equation of a circle in the Cartesian plane is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is its radius. This form is often obtained by completing the square in both x and y for a general quadratic equation in x and y, which is an essential technique in analytic geometry.

Arc Length and Angular Displacement

Arc length in a circle is the distance traveled along its circumference and is directly related to the central angle (in radians) subtended by that arc. The relationship is given by s = r?, where s is the arc length, r is the radius, and ? is the angular displacement. This concept is crucial for converting between linear motion along the circular path and the corresponding change in angle.

Please give Ace some feedback