Kindly help me in finding the differential equation of all circles touching a given straight line at a given point. I think basically, I have to first write the equation of all circles touching a given straight line at a given point, where I have struck. Please help me.
PS: No equation of straight line or the point is given in the question.
Best Answer
Assume the equation of the line is $ax+by=c$ and the point is $(x_0,y_0).$ A vector which gives the direction of the line is $(b,-a).$ Since the line and the circle are tangent at $(x_0,y_0)$ the center is on the line which is perpendicular to the given one. The vector which gives the direction of the perpendicular line is $(a,b).$ Thus, the equation of all possible circles is
$$(x-(x_0\pm ra))^2+(y-(y_0\pm rb))^2=r^2(a^2+b^2)$$
where $r\in (0,\infty).$
We can assume without lost of generality that $a^2+b^2=1.$ In such a case we have that the equations of all possible circles are
$$(x-(x_0\pm ra))^2+(y-(y_0\pm rb))^2=r^2$$
where $r\in (0,\infty)$ is the radius.