Question

Represent a circle with center at (0, 0) and radius 50 mm in parametric form. Hence determine the co-ordinates of a point on this circle at parameter value ‘t’ = 0.6

Answer 1

Answer:

The parametric form of a circle with centre at (0,0) and a radius of 50mm is

                     $x=50cos(t)\ mm\\y=50sin(t)\ mm$

The coordinates of a point where the parameter t=0.6 (in radians) is

                     $x(t=0.6)=41.2\ mm\\y(t=0.6) = 28.23\ mm$

Explanation:

Parametric Equations:

• These are equations that are commonly used to represent coordinates of a geometrical object.
• They are functions of one or more independent variables also called parameters.
• The parametric equation for a circle centred at the point (a,b) and has a radius 'r' is given by

                    $x=a+rcos(\theta)\\y=b+rsin(\theta)$ where $\theta$ is the parameter.

Step 1:

Given the centre of the circle to be (0,0) and the radius $r=50\ mm$.

From the general parametric equation of the circle, we have $a=b=0$ and $\theta=t$.

Step 2:

Putting the values to get the coordinates,

                 $x=50cos(t)\ mm\\y=50sin(t)\ mm$

This is the parametric form for the given circle.

Step 5:

To find it at the parameter value $t=0.6$, we substitute in the above set of equations to find the coordinates.

Let's assume 't' is given in radians. Then,

                 $x=50cos(0.6)\ mm = 41.2\ mm\\y=50sin(0.6)\ mm=28.23\ mm$

Therefore,

The parametric form of a circle with centre at (0,0) and a radius of 50mm is

                     $x=50cos(t)\ mm\\y=50sin(t)\ mm$

The coordinates of a point where the parameter t=0.6 (in radians) is

                     $x(t=0.6)=41.2\ mm\\y(t=0.6) = 28.23\ mm$