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Obtaining Centre and Radius of a Circle from General Equation of a Circle

For what valu...

Question

For what value of $k$ the circles $x^{2} + y^{2} + 5 x + 3 y + 7 = 0$ and $x^{2} + y^{2} - 8 x + 6 y + k = 0$ cut orthogonally.

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Solution

We have,
$x^{2} + y^{2} + 5 x + 3 y + 7 = 0$ $. . . . . . . . . . . (1)$
$x^{2} + y^{2} - 8 x + 6 y + k = 0$ $. . . . . . . . . . . . (2)$

We know that the condition of two circles are orthogonal
$2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$

Therefore,
$2 (\frac{5}{2}) (- 4) + 2 (\frac{3}{2}) (3) = 7 + k$

$- 20 + 9 = 7 + k$

$k = - 18$

Hence, this is the answer.

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