Let a circle passing through (2,0) have its centre at the point (h,k).

Question

Let a circle passing through $(2,0)$ have its centre at the point $(h,k)$. Let $({x}_{c},{y}_{c})$ be the point of intersection of the lines $3x+5y=1$ and $(2+c)x+5{c}^{2}y=1$. If $h=\underset{c	o 1}{lim}{x}_{c}$ and $k=\underset{c	o 1}{lim}{y}_{c}$, then the equation of the circle is :

Accepted Answer

$\begin{gathered} 3x+5y=1 \ (2+c)x+5{c}^{2}y=1 \ 3{c}^{2}x+5{c}^{2}y={c}^{2} \end{gathered}$

Subtracting

$\begin{gathered} (2+c-3{c}^{2})x=1-{c}^{2} \ {x}_{c}=\frac{1-{c}^{2}}{2+c-3{c}^{2}}=\frac{(1-c)(1+c)}{(1-c)(3c+2)}=\frac{c+1}{3c+2} \ y=\frac{1-3x}{5}=\frac{1-3(\frac{c+1}{3c+2})}{5} \ =\frac{3c+2-3c-3}{5(3c+2)} \ {y}_{c}=\frac{-1}{5(3c+2)} \ \underset{c	o 1}{lim}{x}_{c}=\frac{2}{5}=h \ \underset{c	o 1}{lim}{y}_{c}=\frac{-1}{25}=k \end{gathered}$

Equation of circle is

$25{x}^{2}+25{y}^{2}-20x+2y-60=0$

Solutions · 1