The locus of the centre of a circle which cuts a given circle orthogonally and also touches a given straight line is (a) circle (c) parabola (b) line (d) ellipse

Options

A. circle
B. line
C. parabola
D. ellipse

Explanation

To solve the problem of finding the locus of the center of a circle that cuts a given circle orthogonally and also touches a given straight line, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a given circle (let's call it Circle 1) and a straight line (let's call it Line L).
- We need to find the locus of the center of another circle (Circle 2) that intersects Circle 1 orthogonally and touches Line L.

2. **Orthogonal Intersection**:
- For two circles to intersect orthogonally, the relationship between their radii and the distance between their centers must satisfy the equation:
\[ d^2 = r_1^2 + r_2^2 \]
where \(d\) is the distance between the centers of the circles, \(r_1\) is the radius of Circle 1, and \(r_2\) is the radius of Circle 2.

3. **Touching the Line**:
- If Circle 2 touches Line L, the distance from the center of Circle 2 to Line L must equal the radius \(r_2\) of Circle 2.

4. **Setting Up the Coordinate System**:
- Place Circle 1 at the origin (0, 0) with radius \(r_1\) and Line L as the x-axis (y = 0).
- Let the center of Circle 2 be at point \(C(h, k)\) and its radius be \(r_2\).

5. **Using the Conditions**:
- The condition for Circle 2 touching Line L gives us:
\[ k = r_2 \]
- The condition for orthogonal intersection gives:
\[ h^2 + k^2 = r_1^2 + r_2^2 \]

6. **Substituting for k**:
- Substitute \(k = r_2\) into the orthogonality condition:
\[ h^2 + r_2^2 = r_1^2 + r_2^2 \]
- This simplifies to:
\[ h^2 = r_1^2 \]
- Therefore, \(h = \pm r_1\).

7. **Finding the Locus**:
- The locus of the center \(C(h, k)\) can be described by the points \((\pm r_1, r_2)\).
- As \(r_2\) varies, the locus forms a vertical line at \(x = \pm r_1\).

8. **Conclusion**:
- The locus of the center of Circle 2 is a pair of vertical lines at \(x = r_1\) and \(x = -r_1\).

### Final Answer:
The locus of the center of the circle which cuts a given circle orthogonally and also touches a given straight line is a **line**.