Find the locus of the centre of a circle which passes through the origin and cuts off a length `2b` from the line `x=c`.

Options

A. `y^(2)+2cx=b^(2)+c^(2)`
B. `x^(2)+cx=b^(2)+c^(2)`
C. `y^(2)+2cy=b^(2)=b^(2)+c^(2)`
D. `x^(2)+cy=b^(2)+c^(2)`

Answer

To find the locus of the center of a circle that passes through the origin and cuts off a length of \(2b\) from the line \(x = c\), we can follow these steps:

### Step 1: Define the center of the circle
Let the center of the circle be \( (h, k) \).

### Step 2: Identify the radius
Since the circle passes through the origin \( (0, 0) \), the radius \( r \) can be expressed using the distance formula:
\[ r = \sqrt{h^2 + k^2} \]

### Step 3: Determine the points of intersection with the line \( x = c \)
The circle intersects the line \( x = c \) at two points. The coordinates of these points can be expressed as \( (c, y_1) \) and \( (c, y_2) \).

### Step 4: Calculate the distance between the intersection points
The length of the segment cut off by the circle from the line \( x = c \) is given as \( 2b \). Therefore, the distance between the two intersection points is:
\[ |y_1 - y_2| = 2b \]

### Step 5: Use the circle equation
The equation of the circle can be expressed as:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( x = c \) into the circle's equation gives:
\[ (c - h)^2 + (y - k)^2 = r^2 \]

### Step 6: Solve for \( y \)
This can be rearranged to find \( y \):
\[ (y - k)^2 = r^2 - (c - h)^2 \]
Taking the square root gives:
\[ y - k = \pm \sqrt{r^2 - (c - h)^2} \]
Thus, the two intersection points are:
\[ y_1 = k + \sqrt{r^2 - (c - h)^2} \]
\[ y_2 = k - \sqrt{r^2 - (c - h)^2} \]

### Step 7: Set up the equation for the length
The length \( |y_1 - y_2| \) can be expressed as:
\[ |y_1 - y_2| = |(k + \sqrt{r^2 - (c - h)^2}) - (k - \sqrt{r^2 - (c - h)^2})| = 2\sqrt{r^2 - (c - h)^2} \]
Setting this equal to \( 2b \):
\[ 2\sqrt{r^2 - (c - h)^2} = 2b \]
Dividing both sides by 2:
\[ \sqrt{r^2 - (c - h)^2} = b \]

### Step 8: Square both sides
Squaring both sides gives:
\[ r^2 - (c - h)^2 = b^2 \]

### Step 9: Substitute for \( r^2 \)
Substituting \( r^2 = h^2 + k^2 \) into the equation:
\[ h^2 + k^2 - (c - h)^2 = b^2 \]

### Step 10: Expand and simplify
Expanding \( (c - h)^2 \):
\[ h^2 + k^2 - (c^2 - 2ch + h^2) = b^2 \]
This simplifies to:
\[ k^2 + 2ch - c^2 = b^2 \]
Rearranging gives:
\[ k^2 + 2ch = b^2 + c^2 \]

### Step 11: Final form of the locus
The final equation representing the locus of the center \( (h, k) \) is:
\[ k^2 + 2ch = b^2 + c^2 \]