Need help to understand a math task about algebraic and parametric equations

Question

Can anybody please explain this for me?:

Find the algebraic and parametric equations of the circle with centre (-2,3) that passes through (1,-1)

How do I find the algebraic and parametric equations of the circle?

Answer 1

The equation of a circle (algebraically) is given as

$$(x-a)^2+(y-b)^2=r^2$$

where $a$ is the x coordinate of its centre, $b$ is the y coordinate of its centre and r is its radius.

Once you have input these values you can then convert this to a parametric form by simply substituting $x=r{sin}(t)$, $y=r{cos}(t)$.

Answer 2

Verify radius (5= R) with your work. Use standard form to start, then use $\sin, \cos $ for the two parts.

$$ R=5;\, (h,k)= (-2,3) \, ;$$

$$ x = h + R \cos\theta,\,y = k + R \sin\theta,\; 0< \theta< 2 \pi. $$

Please note there can be several parametric representations of the same algebraic equation as it is not unique, like:

$$ x = (h + R) \cos^2 \beta+ (h - R) \sin^2 \beta;,\,y = k+ R \cos \beta \sin \beta \,;\; 0< \beta< \pi. $$