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  • Find the locus of the centres of the circles , where ' a

Find the locus of the centres of the circles \mathrm{x^2+y^2-2 a x-2 b y+2=0}, where ' a ' and ' b ' are parameters, if the tangents from the origin to each of the circles are orthogonal.

Option: 1

x^2+y^2=4


Option: 2

x^2+y^2=2


Option: 3

(x-1)^2+y^2=4


Option: 4

(x+1)^2+y^2=2


Answers (1)

best_answer

The given circle is; \mathrm{x^2+y^2-2 a x-2 b y+2=0 }
or
\mathrm{(x-a)^2+(y-b)^2=a^2+b^2-2 }
it's director circle is \mathrm{ (x-a)^2+(y-b)^2=2\left(a^2+b^2-2\right) }
Given that tangents drawn from the origin to the circle are othrogonal, it implies that director circle of the circle must pass through the origin,
\mathrm{ \begin{array}{ll} \Rightarrow & a^2+b^2=2\left(a^2+b^2-2\right) \\ \Rightarrow & a^2+b^2=4 \end{array} }
Thus the locus of the center of the given circle is; \mathrm{ x^2+y^2=4 }

Posted by

Pankaj Sanodiya

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