The centre C of a variable circle passing through a fixed point `(a,0), a gt 0`, touches the line y=x. Locus of C is a parabola whose directrix is

Options

A. `x+y=0`
B. `x+y=a`
C. `x-y=0`
D. `x-y=a`

Answer

To find the locus of the center \( C \) of a variable circle that passes through the fixed point \( (a, 0) \) and touches the line \( y = x \), we can follow these steps:

### Step 1: Understand the Circle's Properties
The center \( C(h, k) \) of the circle must satisfy two conditions:
1. The circle passes through the point \( (a, 0) \).
2. The distance from the center \( C \) to the line \( y = x \) is equal to the radius of the circle.

### Step 2: Equation of the Circle
The general equation of a circle with center \( C(h, k) \) and radius \( r \) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Since the circle passes through the point \( (a, 0) \), we can substitute this point into the equation:
\[ (a - h)^2 + (0 - k)^2 = r^2 \]
This simplifies to:
\[ (a - h)^2 + k^2 = r^2 \tag{1} \]

### Step 3: Distance from Center to Line
The distance \( d \) from the point \( (h, k) \) to the line \( y = x \) can be calculated using the formula:
\[ d = \frac{|h - k|}{\sqrt{2}} \]
Since this distance is equal to the radius \( r \), we have:
\[ \frac{|h - k|}{\sqrt{2}} = r \tag{2} \]

### Step 4: Equating the Two Conditions
From equations (1) and (2), we can express \( r \) in terms of \( h \) and \( k \):
From (1):
\[ r^2 = (a - h)^2 + k^2 \]
From (2):
\[ r = \frac{|h - k|}{\sqrt{2}} \implies r^2 = \frac{(h - k)^2}{2} \]
Equating both expressions for \( r^2 \):
\[ (a - h)^2 + k^2 = \frac{(h - k)^2}{2} \]

### Step 5: Rearranging the Equation
Multiply through by 2 to eliminate the fraction:
\[ 2((a - h)^2 + k^2) = (h - k)^2 \]
Expanding both sides:
\[ 2(a^2 - 2ah + h^2 + k^2) = h^2 - 2hk + k^2 \]
This simplifies to:
\[ 2a^2 - 4ah + 2h^2 + 2k^2 = h^2 - 2hk + k^2 \]
Rearranging gives:
\[ h^2 + k^2 - 2hk + 4ah - 2a^2 = 0 \]

### Step 6: Completing the Square
To express this in a standard form, we can complete the square for \( h \) and \( k \):
\[ (h - k)^2 + 4ah - 2a^2 = 0 \]
This represents a parabola.

### Conclusion
The locus of the center \( C(h, k) \) of the circle is a parabola with a directrix. The directrix can be determined from the equation derived above.