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Find the locus of the centers of the circles `x^2+y^2-2x-2b y+2=0`, where `a`and `b`are parameters, if the tangents from the origin to each of the circlesare orthogonal. |
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Answer» The given circle is `x^(2)+y^(2)-2ax-2by+2=0` or `(x-a)^(2)+(y-b)^(2)=a^(2)+b^(2)-2` Its director circle is `(x-a)^(2)+(y-b)^(2)=2(a^(2)+b^(2)-2)` Given that tangents drawn from the origin to the circle are orthogonal. It implies that the director circle of the circle must pass through the origin, i.e., `a^(2)+b^(2)=2(a^(2)+b^(2)-2)` or `a^(2)+b^(2)=4` Thus, the locus of the center of the given circle is `x^(2)+y^(2)=4.` |
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