Find the equation of a circle which passes through the point `(2,0)` and whose centre is the limit of the point of intersection of eth lines `3 x+5 y=1` and `(2+alpha) x+5 alpha^{2} y=1`

Explanation

Solving the given equation of the lines `3 x+5 y=1` and `(2+alpha) x+5 alpha^{2} y=1`


We get `x=frac{alpha+1}{3 alpha+2}=frac{1+1}{3+2}=frac{2}{5}` as `alpha rightarrow 1`


Thus, `3 times frac{2}{5}+5 y=1 Rightarrow 5 y=1-frac{6}{5}=frac{-1}{5} Rightarrow y=frac{-1}{25}`


Henve the center of the circle is `(frac{2}{5},-frac{1}{25}right)`


If `r` be the radius of the circle, then its equation is



`(2-frac{2}{5})^{2}+(0+frac{1}{25})^{2}=r^{2}`

` Rightarrow r^{2}=frac{1601}{625}`



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