Q.(a) Find the equation of the circle passing through the points $(1,0)$, $(-1,0)$ and $(0,1)$.

(a) The circle through $(1,0),(-1,0),(0,1)$ is $x^2+y^2=1$. (b) Partial fractions: $-\tfrac{4}{9(x+2)}+\tfrac{4}{9(x-1)}-\tfrac{1}{3(x-1)^2}$.

(a) Circle through $(1,0),(-1,0),(0,1)$

$x^{2}+y^{2}+2gx+2fy+c=0.$

• $(1,0):1+2g+c=0.$
• $(-1,0):1-2g+c=0.$
• $(0,1):1+2f+c=0.$ Subtract first two: $4g=0\Rightarrow g=0$, then $c=-1$; third: $1+2f-1=0\Rightarrow f=0.$ $x^{2}+y^{2}-1=0\Rightarrow x^{2}+y^{2}=1.$

(b) Partial fractions of $\dfrac{x-2}{(x+2)(x-1)^{2}}$

$\frac{x-2}{(x+2)(x-1)^{2}}=\frac{A}{x+2}+\frac{B}{x-1}+\frac{C}{(x-1)^{2}},$

so $x-2=A(x-1)^{2}+B(x+2)(x-1)+C(x+2).$

• $x=1:\ -1=3C\Rightarrow C=-\tfrac13.$ …