A circle touches y-axis at (0,3) and makes an intercept of 2 units on the positive x-axis. Intercept made by the circle on the line `sqrt10 x-3 y=1` in units is

Options

A. 3
B. 6
C. `2sqrt10`
D. `10.`

Answer

To solve the problem step by step, we need to find the intercept made by the circle on the line \(\sqrt{10}x - 3y + 1 = 0\).

### Step 1: Determine the center and radius of the circle
The circle touches the y-axis at the point (0, 3). This means the center of the circle must be at some point \((h, 3)\) where \(h\) is the radius of the circle. Since the circle makes an intercept of 2 units on the positive x-axis, the x-intercepts of the circle will be at points \((h - 1, 0)\) and \((h + 1, 0)\). Therefore, the distance between these two points is 2, confirming that the radius \(r = 1\).

### Step 2: Find the value of \(h\)
Since the circle touches the y-axis at (0, 3), the distance from the center \((h, 3)\) to the y-axis (which is \(h\)) must equal the radius \(r\). Thus, we have:
\[ h = r = 1 \]

### Step 3: Write the equation of the circle
The center of the circle is \((1, 3)\) and the radius is 1. The equation of the circle can be written as:
\[ (x - 1)^2 + (y - 3)^2 = 1 \]

### Step 4: Find the intercepts on the line \(\sqrt{10}x - 3y + 1 = 0\)
To find the intercepts made by the circle on the line, we need to substitute the equation of the line into the equation of the circle. Rearranging the line equation gives:
\[ y = \frac{\sqrt{10}}{3}x + \frac{1}{3} \]

### Step 5: Substitute the line equation into the circle equation
Substituting \(y\) from the line equation into the circle equation:
\[ (x - 1)^2 + \left(\frac{\sqrt{10}}{3}x + \frac{1}{3} - 3\right)^2 = 1 \]
Simplifying the second term:
\[ \frac{\sqrt{10}}{3}x + \frac{1}{3} - 3 = \frac{\sqrt{10}}{3}x - \frac{8}{3} \]
Thus, the equation becomes:
\[ (x - 1)^2 + \left(\frac{\sqrt{10}}{3}x - \frac{8}{3}\right)^2 = 1 \]

### Step 6: Solve the equation
Expanding both sides and simplifying will yield a quadratic equation in \(x\). Solving this quadratic will give us the x-coordinates of the intercepts.

### Step 7: Calculate the distance between the intercepts
The distance between the x-coordinates found will give us the length of the intercept made by the circle on the line.

### Final Answer
After calculating the above steps, the intercept made by the circle on the line \(\sqrt{10}x - 3y + 1 = 0\) is \(2\sqrt{10}\).

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