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Find the angle between the circles given...

Find the angle between the circles given by the equations.
`x^2 + y^2 - 12x - 6y + 41 = 0,`
` x^2 + y^2 + 4x + 6y - 59 = 0.`

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To find the angle between the two circles given by the equations: 1. **Circle 1:** \( x^2 + y^2 - 12x - 6y + 41 = 0 \) 2. **Circle 2:** \( x^2 + y^2 + 4x + 6y - 59 = 0 \) we will follow these steps: ### Step 1: Rewrite the equations in standard form We start by rewriting the equations of the circles in standard form \((x - h)^2 + (y - k)^2 = r^2\). **For Circle 1:** \[ x^2 + y^2 - 12x - 6y + 41 = 0 \] Rearranging gives: \[ x^2 - 12x + y^2 - 6y = -41 \] Completing the square for \(x\) and \(y\): \[ (x - 6)^2 - 36 + (y - 3)^2 - 9 = -41 \] \[ (x - 6)^2 + (y - 3)^2 = 4 \] Thus, the center \(C_1\) is \((6, 3)\) and the radius \(R_1 = 2\). **For Circle 2:** \[ x^2 + y^2 + 4x + 6y - 59 = 0 \] Rearranging gives: \[ x^2 + 4x + y^2 + 6y = 59 \] Completing the square for \(x\) and \(y\): \[ (x + 2)^2 - 4 + (y + 3)^2 - 9 = 59 \] \[ (x + 2)^2 + (y + 3)^2 = 72 \] Thus, the center \(C_2\) is \((-2, -3)\) and the radius \(R_2 = 6\sqrt{2}\). ### Step 2: Find the distance between the centers Now, we calculate the distance \(d\) between the centers \(C_1\) and \(C_2\): \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates: \[ d = \sqrt{((-2) - 6)^2 + ((-3) - 3)^2} \] \[ d = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \] ### Step 3: Use the angle formula The angle \(\theta\) between the two circles can be found using the formula: \[ \cos \theta = \frac{d^2 - R_1^2 - R_2^2}{2R_1R_2} \] Substituting the values: \[ \cos \theta = \frac{10^2 - 2^2 - (6\sqrt{2})^2}{2 \cdot 2 \cdot 6\sqrt{2}} \] Calculating: \[ \cos \theta = \frac{100 - 4 - 72}{24\sqrt{2}} = \frac{24}{24\sqrt{2}} = \frac{1}{\sqrt{2}} \] ### Step 4: Find the angle Now, we find \(\theta\): \[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = 45^\circ \quad \text{or} \quad \frac{\pi}{4} \text{ radians} \] ### Final Answer The angle between the two circles is \(45^\circ\) or \(\frac{\pi}{4}\) radians. ---
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