Length of intercept made by the circle `x^(2)+y^(2)-16x+4y-36=0` on x-axis is

Options

A. 20
B. 10
C. 5
D. none

Answer

To find the length of the intercept made by the circle \( x^2 + y^2 - 16x + 4y - 36 = 0 \) on the x-axis, we will follow these steps:

### Step 1: Rewrite the equation of the circle in standard form
The given equation of the circle is:
\[ x^2 + y^2 - 16x + 4y - 36 = 0 \]
We can rearrange this equation to group the \(x\) and \(y\) terms:
\[ x^2 - 16x + y^2 + 4y = 36 \]

### Step 2: Complete the square for \(x\) and \(y\)
To complete the square for \(x\):
\[ x^2 - 16x = (x - 8)^2 - 64 \]
To complete the square for \(y\):
\[ y^2 + 4y = (y + 2)^2 - 4 \]

Substituting these back into the equation gives:
\[ (x - 8)^2 - 64 + (y + 2)^2 - 4 = 36 \]
Simplifying this, we have:
\[ (x - 8)^2 + (y + 2)^2 - 68 = 36 \]
\[ (x - 8)^2 + (y + 2)^2 = 104 \]

### Step 3: Identify the center and radius of the circle
From the equation \( (x - 8)^2 + (y + 2)^2 = 104 \), we can identify:
- Center \(C(8, -2)\)
- Radius \(r = \sqrt{104} = 2\sqrt{26}\)

### Step 4: Find the y-coordinate of the circle at the x-axis
The x-axis is defined by \(y = 0\). To find the intercepts on the x-axis, we set \(y = 0\) in the equation of the circle:
\[ (x - 8)^2 + (0 + 2)^2 = 104 \]
This simplifies to:
\[ (x - 8)^2 + 4 = 104 \]
\[ (x - 8)^2 = 100 \]
Taking the square root of both sides gives:
\[ x - 8 = 10 \quad \text{or} \quad x - 8 = -10 \]
Thus,
\[ x = 18 \quad \text{or} \quad x = -2 \]

### Step 5: Calculate the length of the intercept
The x-intercepts are at \(x = 18\) and \(x = -2\). The length of the intercept is:
\[ \text{Length} = 18 - (-2) = 18 + 2 = 20 \]

### Final Answer
The length of the intercept made by the circle on the x-axis is \(20\).

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