A variable circle is drawn to touch the line `3x – 4y = 10` and also the circle `x^2 + y^2 = 1` externally then the locus of its centre is -

Options

A. straight line
B. circle
C. pair of real, distinct straight lines
D. parabola

Answer

To find the locus of the center of a variable circle that touches the line \(3x - 4y = 10\) and the circle \(x^2 + y^2 = 1\) externally, we can follow these steps:

### Step 1: Identify the given elements
- The line equation is \(3x - 4y = 10\).
- The circle equation is \(x^2 + y^2 = 1\).
- The center of the circle we are looking for is denoted as \(C(h, k)\).

### Step 2: Calculate the distance from the center to the line
The distance \(d\) from a point \((h, k)\) to the line \(Ax + By + C = 0\) is given by the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
For the line \(3x - 4y - 10 = 0\), we have:
- \(A = 3\)
- \(B = -4\)
- \(C = -10\)

Thus, the distance from the center \(C(h, k)\) to the line is:
\[ d = \frac{|3h - 4k - 10|}{\sqrt{3^2 + (-4)^2}} = \frac{|3h - 4k - 10|}{5} \]

### Step 3: Set the distance equal to the radius
Let the radius of the variable circle be \(r\). Since the circle touches the line, the distance from the center to the line must equal the radius:
\[ \frac{|3h - 4k - 10|}{5} = r \quad \text{(1)} \]

### Step 4: Calculate the distance from the center to the circle
The distance from the center \(C(h, k)\) to the center of the circle \(O(0, 0)\) is:
\[ \sqrt{h^2 + k^2} \]
Since the variable circle touches the circle \(x^2 + y^2 = 1\) externally, the distance from the center \(C(h, k)\) to the origin must equal the radius plus the radius of the circle:
\[ \sqrt{h^2 + k^2} = r + 1 \quad \text{(2)} \]

### Step 5: Equate the two expressions for \(r\)
From equation (1), we can express \(r\):
\[ r = \frac{|3h - 4k - 10|}{5} \]
Substituting this into equation (2):
\[ \sqrt{h^2 + k^2} = \frac{|3h - 4k - 10|}{5} + 1 \]

### Step 6: Rearranging the equation
Multiply through by 5 to eliminate the fraction:
\[ 5\sqrt{h^2 + k^2} = |3h - 4k - 10| + 5 \]

### Step 7: Squaring both sides
Square both sides to eliminate the square root:
\[ 25(h^2 + k^2) = (3h - 4k - 10)^2 + 10(3h - 4k - 10) + 25 \]

### Step 8: Simplifying the equation
This will lead to a quadratic equation in \(h\) and \(k\) which represents the locus of the center of the variable circle.

### Conclusion
The locus of the center \(C(h, k)\) of the variable circle is a parabola.