If the equation `x cos theta +y sin theta=p` represents the equation of common chord APQB of the circles `x^(2)+y^(2)=a^(2) ` and `x^(2)+y^(2)=b^(2)(a gt b)` then AP =

Options

A. `sqrt(a^(2)+p^(2)) +sqrt(b^(2)+p^(2))`
B. `sqrt(a^(2)+p^(2))-sqrt(b^(2)+p^(2))`
C. `sqrt(a^(2)-p^(2))+sqrt(b^(2)-p^(2))`
D. `sqrt(a^(2)-p^(2))-sqrt(b^(2)-p^(2))`

Answer

To solve the problem, we need to find the length of segment AP, which is part of the common chord of two circles defined by the equations \(x^2 + y^2 = a^2\) and \(x^2 + y^2 = b^2\) (where \(a > b\)). The equation of the common chord is given by \(x \cos \theta + y \sin \theta = p\).

### Step-by-Step Solution

1. **Identify the Centers and Radii of the Circles:**
- The center of both circles is at the origin \(O(0, 0)\).
- The radius of the first circle (larger) is \(OA = a\).
- The radius of the second circle (smaller) is \(OP = b\).

2. **Understand the Equation of the Common Chord:**
- The equation \(x \cos \theta + y \sin \theta = p\) represents a line, which is the common chord of the two circles.

3. **Determine the Perpendicular Distance from the Origin to the Line:**
- The distance from the origin \(O\) to the line \(x \cos \theta + y \sin \theta = p\) is given as \(OL = p\).

4. **Calculate Length AL:**
- In triangle \(ALO\), we can use the Pythagorean theorem:
\[ AL^2 = OA^2 - OL^2 \]
Substituting the known values:
\[ AL^2 = a^2 - p^2 \]
Thus, we find:
\[ AL = \sqrt{a^2 - p^2} \]

5. **Calculate Length PL:**
- In triangle \(OPL\), we again use the Pythagorean theorem:
\[ PL^2 = OP^2 - OL^2 \]
Substituting the known values:
\[ PL^2 = b^2 - p^2 \]
Thus, we find:
\[ PL = \sqrt{b^2 - p^2} \]

6. **Calculate Length AP:**
- The length of segment \(AP\) can now be calculated as:
\[ AP = AL - PL \]
Substituting the expressions we found:
\[ AP = \sqrt{a^2 - p^2} - \sqrt{b^2 - p^2} \]

### Final Result
Thus, the length of segment \(AP\) is:
\[ AP = \sqrt{a^2 - p^2} - \sqrt{b^2 - p^2} \]