Geo4.37

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Find the equations of circles which touch the axis of x at the origin and the line 4x-3y+24=0\,

Let the equation of the required circle is

x^{2}+y^{2}+2gx+2fy+c=0\,

Let the equation to the tangent at the origin (0,0) to the above be gx+fy+c=0

Given the tangent at(0,0) is x axis i.e y=0,means g=0,c=0

From the first one and just the above one,we have

x^{2}+y^{2}+2fy=0\,

Given that this equation touches the line 4x-3y+24=0\,

Perpendicular distance from (0,-f) on the above line is radius |f|.

|{\frac  {4(0)-3(-f)+24}{{\sqrt  {16+9}}}}|=|f|\,

Hence

3f+24=\pm 5f\,

This gives

f=12,-3\,

The equations of the required circle are

x^{2}+y^{2}+24y=0,x^{2}+y^{2}-6y=0\,


Main Page:Geometry:Circles