The circle $ax^{2} + ay^{2} + 2g_{1}x + 2f_{1}y + c_{1} = 0$ and $bx^{2} + by^{2} + 2g_{2}x + 2f_{2}y + c_{2} = 0$ ($a
eq 0$ and $b
eq 0$) cut orthogonally, if

Question

The circle $ax^{2} + ay^{2} + 2g_{1}x + 2f_{1}y + c_{1} = 0$ and $bx^{2} + by^{2} + 2g_{2}x + 2f_{2}y + c_{2} = 0$ ($a 
eq 0$ and $b 
eq 0$) cut orthogonally, if

Dr. Abubacker · Accepted Answer

(b)
Given circles can be rewritten as
$x^{2} + y^{2} + \frac{2g_{1}}{a} + \frac{2f_{1}}{a}y + \frac{c_{1}}{a} = 0$
And $x^{2} + y^{2} + \frac{2g_{2}}{b}x + \frac{2f_{2}}{b}y + \frac{c_{2}}{b} = 0$
Centres of circles are $C_{1}\left( - \frac{g_{1}}{a},\  - \frac{f_{1}}{a} ight)$ and $C_{2}\left( - \frac{g_{2}}{b},\  - \frac{f_{2}}{b} ight)$
respectively,
We know, if two circles cut orthogonally, then
$2\left( G_{1} + G_{2} + F_{1}F_{2} ight) = C_{1} + C_{2}$
$	herefore\ \ 2\left( \frac{g_{1}g_{2}}{ab} + \frac{f_{1}f_{2}}{ab} ight) = \frac{c_{1}}{a} + \frac{c_{2}}{b}$
$\Rightarrow \ \ 2\left( g_{1}g_{2} + f_{1}f_{2} ight) = bc_{1} + ac_{2}$

Q. The circle $ax^{2} + ay^{2} + 2g_{1}x + 2f_{1}y + c_{1} = 0$ and $bx^{2} + by^{2} + 2g_{2}x + 2f_{2}y + c_{2} = 0$ ($a \neq 0$ and $b \neq 0$) cut orthogonally, if

Explanation

Reviewed by Dr. Abubacker

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