Can someone please be kind and show me the steps on this? Thanks
Find the equation of the circle centered at the origin in the uv-plane that has twice the circumference of the circle whose equation equals
(u-5)^2 + y^2 =9
The general equation of a circle is
(x-x0)² + (y-y0)² = r²
where (x0,y0) represent the centre of the circle, and r = radius.
The given equation of the circle:
(u-5)^2 + y^2 =9
can be rewritten as:
(u-5)² + (y-0)² = 3²
which tells us that the circle has a centre at (5,0) with a radius of 3.
The circumference is therefore 2π(3)=6π
The new circle is to have twice the circumference, 12π so the radius must be 6.
The centre is at the origin, i.e. (0,0).
So the equation of the required circle is therefore:
(u-0)²+(v-0)²=6²
Simplifying,
u²+v²=36
Thank you very, very much Mathmate for your help I appreciate the time you took to show me the step by step...Again many thanks
Sure! I can help you find the equation of the circle centered at the origin in the uv-plane.
First, let's start by finding the circumference of the circle whose equation is given as (u-5)^2 + y^2 = 9.
The equation of a circle can be written in standard form as (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r is the radius.
In this case, since the circle is centered at (5, 0) and has a radius of 3 (sqrt(9)), we can rewrite the equation as (u - 5)^2 + y^2 = 3^2.
Now, let's find the circumference of this circle. The circumference of a circle can be found using the formula C = 2πr, where r is the radius.
Considering the given circle has a radius of 3, we can calculate its circumference as C = 2 * π * 3 = 6π.
The question asks for a circle in the uv-plane that has twice the circumference of the given circle. So, we need to find a circle with a circumference of 2 * 6π = 12π.
Next, let's find the equation of this new circle in the uv-plane, centered at the origin.
Since the circle is centered at the origin, the equation can be written in the form u^2 + v^2 = r^2, where (0, 0) represents the center and r is the radius.
To find the radius, we know that the circumference of this circle is 12π. Using the formula C = 2πr, we can solve for r:
12π = 2πr
Dividing both sides by 2π:
r = 6
Therefore, the equation of the circle in the uv-plane, centered at the origin, is:
u^2 + v^2 = 6^2
u^2 + v^2 = 36.
That's it! The equation of the circle in the uv-plane, centered at the origin, is u^2 + v^2 = 36.