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If the intercepts of the variable circle...

If the intercepts of the variable circle on the x- and yl-axis are 2 units and 4 units, respectively, then find the locus of the center of the variable circle.

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Let the equation of variable circle be `x^(2)+y^(2)+2gx+2fy+c=0`
We have to find the locus of the centre C( -g,-f).
According to the question, we have
`2sqrt(g^(2)-c)=2` and `2 sqrt(f^(2)-c)=4`
`:. g^(2)-c=1` ad `f^(2)-c=4`
Eliminating c, we gt
`f^(2)-g^(2)=3` or `(-f)^(2)-(-g)^(2)=3`
Therefore, required locus is `y^(2)-x^(2)=3`.
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