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Q.
The radius of the tangent circle that can be drawn to pass through the points $(0,7)$ and $(0,6)$ and touching the $x$-axis is
Conic Sections
Solution:
The equation of any circle through $(0,1)$ and $(0,6)$ is
$x^{2}+(y-1)(y-6)+\lambda x=0 $
or $x^{2}+y^{2}+\lambda x-7 y+6=0$
If it touches the $x$-axis, then $x^{2}+\lambda x+6=0$
should have equal roots, i.e.,
$\lambda^{2}=24 $ or $ \lambda=\pm \sqrt{24}$
Radius of these circles $=\sqrt{6+\frac{49}{4}-6}$
$=\frac{7}{2}$ units
So, we can draw two circles but the radius of each circle is $7 / 2$.