Geo4.45

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Find the equation of the circle with centre on the line 2x+y=0 and which touches the lines 4x-3y+10=0 and 4x-3y-30=0.


Let the required circle be

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

The centre is(-g,-f) and radius is {\sqrt  {g^{2}+f^{2}-c}}\,

The centre lies on 2x+y=0\,

Hence

2g+f=0,f=-2g\,

The distance from the centre to the first line =radius

|{\frac  {-4g+3f+10}{5}}|={\sqrt  {g^{2}+f^{2}-c}}\,

Squaring on both sides,we get

(-4g+3f+10)=25(g^{2}+f^{2}-c)\,

Now

100(g-1)^{2}=25(5g^{2}-c)\,

4g^{2}-8g+4=5g^{2}-c\,

g^{2}+8g-c=4\,

Similarly for the second line, we have

|{\frac  {-4g+3f-30}{5}}|={\sqrt  {g^{2}+f^{2}-c}}\,

(-4g+3f-30)^{2}=25(g^{2}+f^{2}-c)\,

100(g+3)^{2}=25(5g^{2}-c)\,

4g^{2}+24g+36=5g^{2}-c\,

g^{2}-24g-c=36\,

Substracting these two equations, we have

32g=-32,g=-1\,

1^{2}+24-c=36,c-11\,

f=2\,

The equation of the required circle is

x^{2}+y^{2}-2x+4y-11=0\,


Main Page:Geometry:Circles