The center of a sphere which touches the lines `y=x,z=c and y=-x,z=-c` lies on

Options

A. `xy+2cz=0`
B. `yz+2cx=0`
C. `zx+2cy=0`
D. none

Answer

To find the locus of the center of a sphere that touches the lines \( y = x, z = c \) and \( y = -x, z = -c \), we can follow these steps:

### Step 1: Understand the Sphere's Equation
The general equation of a sphere can be expressed as:
\[ x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0 \]
where \((u, v, w)\) are the coordinates of the center of the sphere.

### Step 2: Analyze the First Line
The first line is defined by \( y = x \) and \( z = c \). We substitute these into the sphere's equation:
\[ x^2 + (x)^2 + (c)^2 + 2ux + 2vx + 2wc + d = 0 \]
This simplifies to:
\[ 2x^2 + c^2 + 2(2u + v)x + 2wc + d = 0 \]

### Step 3: Condition for Tangency
For the sphere to touch the line, the quadratic equation in \( x \) must have a double root. Therefore, the discriminant must be zero:
\[ (2(2u + v))^2 - 4 \cdot 2 \cdot (c^2 + 2wc + d) = 0 \]
This leads to:
\[ 4(2u + v)^2 - 8(c^2 + 2wc + d) = 0 \]
Dividing by 4 gives:
\[ (2u + v)^2 - 2(c^2 + 2wc + d) = 0 \]

### Step 4: Analyze the Second Line
Now we consider the second line defined by \( y = -x \) and \( z = -c \). Substituting these values into the sphere's equation gives:
\[ x^2 + (-x)^2 + (-c)^2 + 2ux + 2v(-x) + 2w(-c) + d = 0 \]
This simplifies to:
\[ 2x^2 + c^2 + 2(2u - v)x - 2wc + d = 0 \]

### Step 5: Condition for Tangency of the Second Line
Similarly, for the sphere to touch this line, we set the discriminant to zero:
\[ (2(2u - v))^2 - 4 \cdot 2 \cdot (c^2 - 2wc + d) = 0 \]
This leads to:
\[ 4(2u - v)^2 - 8(c^2 - 2wc + d) = 0 \]
Dividing by 4 gives:
\[ (2u - v)^2 - 2(c^2 - 2wc + d) = 0 \]

### Step 6: Subtract the Two Conditions
Now we have two equations:
1. \((2u + v)^2 = 2(c^2 + 2wc + d)\)
2. \((2u - v)^2 = 2(c^2 - 2wc + d)\)

Subtracting these two equations:
\[ (2u + v)^2 - (2u - v)^2 = 2(c^2 + 2wc + d) - 2(c^2 - 2wc + d) \]
This simplifies to:
\[ 4uv = 8wc \]
Thus,
\[ uv = 2wc \]

### Step 7: Locus of the Center
The locus of the center \((u, v, w)\) can be expressed as:
\[ uv - 2wc = 0 \]
Rearranging gives:
\[ uv + 2cz = 0 \]
where \( z = w \).

### Final Result
Thus, the locus of the center of the sphere is given by:
\[ xy + 2cz = 0 \]