Find the equation of sphere which passes through `(1, 0, 0)` and has its centre on the positive direction of Y-axis and has radius 2.

Answer

To find the equation of the sphere that passes through the point (1, 0, 0), has its center on the positive direction of the Y-axis, and has a radius of 2, we can follow these steps:

### Step 1: Identify the center of the sphere
Since the center of the sphere is on the positive Y-axis, we can denote the center as \( (0, y_0, 0) \), where \( y_0 > 0 \).

### Step 2: Use the radius condition
The radius of the sphere is given as 2. The distance from the center of the sphere to the point (1, 0, 0) must equal the radius. Therefore, we can set up the equation for the distance:

\[ \text{Distance} = \sqrt{(1 - 0)^2 + (0 - y_0)^2 + (0 - 0)^2} = 2 \]

This simplifies to:

\[ \sqrt{1 + y_0^2} = 2 \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives us:

\[ 1 + y_0^2 = 4 \]

### Step 4: Solve for \( y_0^2 \)
Rearranging the equation, we find:

\[ y_0^2 = 4 - 1 = 3 \]

### Step 5: Find \( y_0 \)
Taking the positive square root (since \( y_0 \) is on the positive Y-axis):

\[ y_0 = \sqrt{3} \]

### Step 6: Write the center of the sphere
Now we have the center of the sphere as \( (0, \sqrt{3}, 0) \).

### Step 7: Write the equation of the sphere
The general equation of a sphere with center \( (h, k, l) \) and radius \( r \) is given by:

\[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]

Substituting \( h = 0 \), \( k = \sqrt{3} \), \( l = 0 \), and \( r = 2 \):

\[ (x - 0)^2 + (y - \sqrt{3})^2 + (z - 0)^2 = 2^2 \]

This simplifies to:

\[ x^2 + (y - \sqrt{3})^2 + z^2 = 4 \]

### Step 8: Expand the equation
Expanding the equation gives:

\[ x^2 + (y^2 - 2\sqrt{3}y + 3) + z^2 = 4 \]

Combining like terms results in:

\[ x^2 + y^2 + z^2 - 2\sqrt{3}y + 3 = 4 \]

### Step 9: Rearranging the equation
Finally, we can rearrange this to get the standard form:

\[ x^2 + y^2 + z^2 - 2\sqrt{3}y - 1 = 0 \]

This is the equation of the sphere.

### Final Answer
The equation of the sphere is:

\[ x^2 + y^2 + z^2 - 2\sqrt{3}y - 1 = 0 \]

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