A circle `C_(1)` has radius 2 units and a circles `C_(2)` has radius 3 units. The distance between the centres of `C_(1)` and `C_(2)` is 7 units. If two points, one tangent to both circles and the other passing through the centre of both circles, intersect at point P which lies between the centers of `C_(1) and C_(2)`, then the distance between P and the centre of `C_(2)` is

Options

A. `(9)/(4)" units"`
B. `(7)/(3)" units"`
C. `(21)/(5)" units"`
D. `(14)/(5)" units"`

Answer

To solve the problem, we need to find the distance between point P and the center of circle C2, given the conditions of the circles C1 and C2.

### Step-by-Step Solution:

1. **Identify the centers and radii of the circles:**
- Let the center of circle C1 be O1 and the center of circle C2 be O2.
- The radius of circle C1 (r1) = 2 units.
- The radius of circle C2 (r2) = 3 units.
- The distance between the centers O1 and O2 = 7 units.

2. **Determine the position of the tangent point:**
- The tangent point T lies on the line connecting O1 and O2. Since it is tangent to both circles, we can use the property of tangents.
- The distance from O1 to the tangent point T is equal to the radius of circle C1, which is 2 units.
- The distance from O2 to the tangent point T is equal to the radius of circle C2, which is 3 units.

3. **Calculate the distance from O1 to T and O2 to T:**
- The total distance between O1 and O2 is 7 units.
- Therefore, the distance from O1 to T (d1) + the distance from O2 to T (d2) = 7 units.
- We have d1 = 2 units and d2 = 3 units.
- Thus, d1 + d2 = 2 + 3 = 5 units.

4. **Find the remaining distance between the tangent point and the centers:**
- The remaining distance between the tangent point T and the line connecting O1 and O2 is 7 - 5 = 2 units.
- Since point P lies between the centers O1 and O2, we can place point P at a distance from O2.

5. **Calculate the distance from P to O2:**
- Since the distance from O1 to O2 is 7 units, and point P lies between them, we can express the distance from O2 to P as follows:
- Let the distance from O2 to P be x. Then the distance from O1 to P would be (7 - x).
- Since the tangent point is 2 units from O1, we have:
- Distance from O1 to P = 2 units.
- Therefore, 7 - x = 2.
- Solving for x gives us x = 7 - 2 = 5 units.

### Final Answer:
The distance between point P and the center of circle C2 (O2) is **5 units**.