Circle with centre on a line, and tangents

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Carry out the following:

• (a) write the equation of the circle with centre $A(5;-1)$ and radius $4$;
• (b) write the equation of a circle through $A(5;-1)$ and $B(0;0)$ with centre on the line $y=2x+1$;
• (c) find the tangents to the circle in (a) through $P(-2;2)$.

Solution

(a) $(x-5)^2+(y+1)^2=16$, i.e. $x^2+y^2-10x+2y+10=0$.

(b) Centre $C(t;\,2t+1)$. Imposing $\overline{CO}=\overline{CA}$: $6t=28\Rightarrow t=\tfrac{14}{3}$. Hence $C\left(\tfrac{14}{3};\tfrac{31}{3}\right)$ and $R^2=\tfrac{1157}{9}$: $\left(x-\tfrac{14}{3}\right)^2+\left(y-\tfrac{31}{3}\right)^2=\tfrac{1157}{9}$.

(c) Line $y-2=m(x+2)$. Distance from $C(5;-1)$ equal to $4$: $\dfrac{|7m+3|}{\sqrt{m^2+1}}=4\Rightarrow 33m^2+42m-7=0\Rightarrow m=\dfrac{-21\pm4\sqrt{42}}{33}$.

$\boxed{(x-5)^2+(y+1)^2=16;\quad C\left(\tfrac{14}{3};\tfrac{31}{3}\right);\quad m=\tfrac{-21\pm4\sqrt{42}}{33}}$