Question

The circle passing through the point $(-1,0)$ and touching the $y$-axis at $(0,2)$ also passes through the point:

• A. $(-\frac{3}{2},0)$
• B. $(-\frac{5}{2},2)$
• C. $(3,5)$
• D. $(-4,0)$

Answer 1

The correct option is D $(-4,0)$

Circle touching $y$-axis at $(0,2)$ is $(x-0{)}^2+(y-2{)}^2+\lambda x=0$ passes through $(-1,0)$
$\therefore 1+4-\lambda =0\Rightarrow \lambda =5$
$\therefore x^2+y^2+5x-4y+4=0$
Put value of options in equation of circle, we get
$\therefore$ Circle passes through $(-4,0)$

Alternate Solution:
Now using Pythagoras theorem
$k^2=2^2+(k+1{)}^2\Rightarrow k=-3/2$
Hence circle will pass through $(-4,0)$