In intercept made by the circle with centre (2,3) and radius 6 on y-axis is

Options

A. `18sqrt(2)`
B. `12sqrt(2)`
C. `8sqrt(2)`
D. `6sqrt(2)`

Answer

To find the intercept made by the circle with center (2, 3) and radius 6 on the y-axis, we can follow these steps:

### Step 1: Understand the Circle's Equation
The general equation of a circle with center (h, k) and radius r is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
For our circle, the center is (2, 3) and the radius is 6. Therefore, the equation of the circle is:
\[ (x - 2)^2 + (y - 3)^2 = 6^2 \]
This simplifies to:
\[ (x - 2)^2 + (y - 3)^2 = 36 \]

### Step 2: Find the Points of Intersection with the Y-Axis
To find the intercepts on the y-axis, we set \(x = 0\) in the circle's equation:
\[ (0 - 2)^2 + (y - 3)^2 = 36 \]
This simplifies to:
\[ 4 + (y - 3)^2 = 36 \]
Subtracting 4 from both sides gives:
\[ (y - 3)^2 = 32 \]

### Step 3: Solve for y
Taking the square root of both sides, we have:
\[ y - 3 = \pm \sqrt{32} \]
Since \(\sqrt{32} = 4\sqrt{2}\), we can write:
\[ y - 3 = \pm 4\sqrt{2} \]
Thus, we have two solutions for y:
\[ y = 3 + 4\sqrt{2} \quad \text{and} \quad y = 3 - 4\sqrt{2} \]

### Step 4: Calculate the Length of the Intercept
The y-intercepts are \(y = 3 + 4\sqrt{2}\) and \(y = 3 - 4\sqrt{2}\). The length of the intercept on the y-axis is the distance between these two points:
\[ \text{Length of intercept} = (3 + 4\sqrt{2}) - (3 - 4\sqrt{2}) = 4\sqrt{2} + 4\sqrt{2} = 8\sqrt{2} \]

### Final Answer
Thus, the intercept made by the circle on the y-axis is:
\[ \boxed{8\sqrt{2}} \]

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