When do the two given circles x^2 + y^2 + ax | vedclass

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(A) The condition for two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ to intersect orthogonally is $2g_1g_

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$ad + be = 2(c + f)$

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(A) The condition for two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ to intersect orthogonally is $2g_1g_2 + 2f_1f_2 = c_1 + c_2$.Comparing the given equations with the standard form:$2g_1 = a \implies g_1 = \frac{a}{2}$,$2f_1 = b \implies f_1 = \frac{b}{2}$,$c_1 = c$.$2g_2 = d \implies g_2 = \frac{d}{2}$,$2f_2 = e \implies f_2 = \frac{e}{2}$,$c_2 = f$.Substituting these into the condition:$2(\frac{a}{2})(\frac{d}{2}) + 2(\frac{b}{2})(\frac{e}{2}) = c + f$$\frac{ad}{2} + \frac{be}{2} = c + f$Multiplying by $2$ on both sides:$ad + be = 2(c + f)$.

When do the two given circles $x^2 + y^2 + ax + by + c = 0$ and $x^2 + y^2 + dx + ey + f = 0$ intersect each other orthogonally?

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