For what value of k do the circles x^2 + y^2 | vedclass

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(C) 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 $2(g_1g

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$-18$

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(C) 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 $2(g_1g_2 + f_1f_2) = c_1 + c_2$.Comparing the given equations with the standard form:For the first circle: $2g_1 = 5 \Rightarrow g_1 = 5/2$,$2f_1 = 3 \Rightarrow f_1 = 3/2$,$c_1 = 7$.For the second circle: $2g_2 = -8 \Rightarrow g_2 = -4$,$2f_2 = 6 \Rightarrow f_2 = 3$,$c_2 = k$.Substituting these values into the condition:$2((5/2)(-4) + (3/2)(3)) = 7 + k$$2(-10 + 9/2) = 7 + k$$2(-20/2 + 9/2) = 7 + k$$2(-11/2) = 7 + k$$-11 = 7 + k$$k = -11 - 7 = -18$.

For what value of $k$ do the circles $x^2 + y^2 + 5x + 3y + 7 = 0$ and $x^2 + y^2 - 8x + 6y + k = 0$ intersect orthogonally?

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