First slide

Definition of a circle

Question

The locus of the centre of the circle which cuts the circles $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0$ and $x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ orthogonally, is

Moderate

A

an ellipse
B

the radical axis of the given circles
C

a conic
D

another circle

Solution

Let the circle be $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
This circle cuts the two given circles orthogonally
$∴ 2 (g g_{1} + f f_{1}) = c + c_{1}$
and $2 (g g_{2} + f f_{2}) = c + c_{2}$
Subtracting (ii) from (i), we get
$2 g (g_{1} - g_{2}) + 2 f (f_{1} - f_{2}) = c_{1} - c_{2} .$
$- 2 x (g_{1} - g_{2}) - 2 y (f_{1} - f_{2}) = c_{1} - c_{2}$
$\Rightarrow 2 x (g_{1} - g_{2}) + 2 y (f_{1} - f_{2}) + c_{1} - c_{2} = 0$

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