A variable circle is drawn to touch the line 3x - 4y = 10 and also the circle x² + y² = 1 externally then the locus of its center is:

• (a) Straight line
• (b) Circle
• (c) Pair of real, distinct straight lines
• (d) Parabola

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 need to analyze the conditions for tangency.

Understanding the Line and Circle

The line can be rewritten in slope-intercept form as:

\(y = \frac{3}{4}x - \frac{5}{2}\)

This line has a slope of \(\frac{3}{4}\) and a y-intercept of \(-\frac{5}{2}\).

Circle Characteristics

The given circle \(x^2 + y^2 = 1\) has a center at the origin (0, 0) and a radius of 1.

Circle Touching Conditions

• The distance from the center of the variable circle to the line must equal its radius.
• The distance from the center of the variable circle to the origin must equal the sum of the radii of both circles.

Finding the Locus

Let the center of the variable circle be at point \( (h, k) \) and its radius be \( r \). The distance from the center to the line is given by:

\( \frac{|3h - 4k - 10|}{\sqrt{3^2 + (-4)^2}} = r \)

For the external tangency with the unit circle, the distance from the center to the origin must satisfy:

\( \sqrt{h^2 + k^2} = r + 1 \)

Resulting Locus

By manipulating these equations, we can derive that the locus of the center \( (h, k) \) forms a parabola. Thus, the correct answer is: