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$$C_1+\lambda \,C_2=0$$

are all circles passing through intersection of

$$C_1=0,C_2=0$$

Is it possible to express orthogonal trajectories of all the $C_1+\lambda \,C_2=0 $ circles using another parameter $\mu?$

Attempts with complex algebra and bipolar coordinates have not been lucky.

EDIT 1:

I took circles in the form

$$ \dfrac{x^2+y^2 + 2 f_1 x +2g_1 y +c_1}{x^2+y^2 + 2 f_2 x +2g_2 y +c_2} = -\lambda $$ by differentiating and simplifying got it into form

$$ \dfrac{dy}{dx}=\dfrac{y+k}{x+h}$$

and integrated to get straight lines

$$ (y+k) = \mu\, (x+h)$$

but find this wrong.

  • 0
    One of those $C_1 + \lambda C_2$ circles is (for $\lambda = -1$) a line passing through the intersection of circles. Would that help (at least for intuition)? – 2017-02-19
  • 0
    The line is radical axis known, but what should come next? – 2017-02-19

0 Answers 0