Questions: Find the x - and y-intercepts of the given circle. the circle with center (5,-4) and radius 6

Find the x - and y-intercepts of the given circle. the circle with center (5,-4) and radius 6
Transcript text: Find the $x$ - and $y$-intercepts of the given circle. the circle with center $(5,-4)$ and radius 6
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Solution

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Solution Steps

Step 1: Circle Equation

The equation of the circle with center \((5, -4)\) and radius \(6\) is given by: \[ (x - 5)^2 + (y + 4)^2 = 6^2 \]

Step 2: Finding x-Intercepts

To find the x-intercepts, set \(y = 0\) in the circle equation: \[ (x - 5)^2 + (0 + 4)^2 = 36 \] This simplifies to: \[ (x - 5)^2 + 16 = 36 \] Subtracting \(16\) from both sides gives: \[ (x - 5)^2 = 20 \] Taking the square root of both sides results in: \[ x - 5 = \pm \sqrt{20} \] Thus, the x-intercepts are: \[ x = 5 + \sqrt{20} \quad \text{and} \quad x = 5 - \sqrt{20} \]

Step 3: Finding y-Intercepts

To find the y-intercepts, set \(x = 0\) in the circle equation: \[ (0 - 5)^2 + (y + 4)^2 = 36 \] This simplifies to: \[ 25 + (y + 4)^2 = 36 \] Subtracting \(25\) from both sides gives: \[ (y + 4)^2 = 11 \] Taking the square root of both sides results in: \[ y + 4 = \pm \sqrt{11} \] Thus, the y-intercepts are: \[ y = -4 + \sqrt{11} \quad \text{and} \quad y = -4 - \sqrt{11} \]

Final Answer

The x-intercepts are \( \boxed{5 + \sqrt{20}} \) and \( \boxed{5 - \sqrt{20}} \).
The y-intercepts are \( \boxed{-4 + \sqrt{11}} \) and \( \boxed{-4 - \sqrt{11}} \).

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