Find the equation of a circle passing through the origin and cutting the circles
$x^{2} + y^{2} + 2 g_{1} x + 2 f y_{1} + c_{1} = 0$
and $x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ orthogonally.

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Solution

Let the required circle through origin be
$x^{2} + y^{2} + 2 g x + 2 f y = 0 . . . . . (1)$
Condition of orthogonality gives
$2 g g_{1} + 2 f f_{1} = c_{1} . . . . . (2)$
$2 g g_{2} + 2 f f_{2} = c_{2} . . . . (3)$
Eliminating the unknowns $2 g$ and $2 f$ from $(1), (2)$ and $(3)$ , we get
$∣ ∣ ∣ ∣ ∣ \begin{matrix} x & y & x^{2} + y^{2} g_{1} & f_{1} & - c_{1} g_{2} & f_{2} & - c_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
Above is the required equation of circle.

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Similar questions

Q. Prove that the circle through the origin and cutting 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

∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{2} + y^{2} & - x & - y c_{1} & g_{1} & f_{1} c_{2} & g_{2} & f_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

Two circles $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0 a n d x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c^{2} = 0$ are said to be orthogonal. Then $2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$

Q. Assertion :The equation of the circle passing through the point

(2 a, 0)

and whose radical axis is

x = \frac{a}{2}

with respect to the circle

x^{2} + y^{2} = a^{2}

, will be

x^{2} + y^{2} - 2 a x = 0

Reason: The equation of radical axis of two 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

2 (g_{1} - g_{2}) x + 2 (f_{1} - f_{2}) y + (c_{1} - c_{2}) = 0

The locus of center 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

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Find the equation of a circle passing through the origin and cutting the circlesx2+y2+2g1x+2fy1+c1=0and x2+y2+2g2x+2f2y+c2=0 orthogonally.

Find the equation of a circle passing through the origin and cutting the circles
$x^{2} + y^{2} + 2 g_{1} x + 2 f y_{1} + c_{1} = 0$
and $x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ orthogonally.