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 is equal to `:`

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 the segment AP, which is part of the common chord of the two given circles. Let's break down the solution step by step.

### Step 1: Understand the Circles
The equations of the circles are given as:
1. \( x^2 + y^2 = a^2 \) (Circle with radius \( a \))
2. \( x^2 + y^2 = b^2 \) (Circle with radius \( b \))

Since \( a > b \), these circles are concentric, meaning they share the same center at the origin (0, 0).

### Step 2: Identify the Line Equation
The equation of the common chord is given by:
\[ x \cos \theta + y \sin \theta = p \]
This line has a perpendicular distance \( p \) from the center of the circles (the origin).

### Step 3: Calculate the Length of the Perpendicular from the Center
Let \( OL \) be the perpendicular distance from the center (O) to the line. Thus, we have:
\[ OL = p \]

### Step 4: Calculate Segment AL
The length \( AL \) can be calculated using the Pythagorean theorem in triangle OAL:
\[ AL = \sqrt{OA^2 - OL^2} \]
Where \( OA = a \) (the radius of the larger circle). Therefore:
\[ AL = \sqrt{a^2 - p^2} \]

### Step 5: Calculate Segment PL
Similarly, for the smaller circle, we can calculate \( PL \):
\[ PL = \sqrt{OB^2 - OL^2} \]
Where \( OB = b \) (the radius of the smaller circle). Thus:
\[ PL = \sqrt{b^2 - p^2} \]

### Step 6: Calculate AP
The length of the segment \( AP \) is given by the difference between \( AL \) and \( PL \):
\[ AP = AL - PL \]
Substituting the values we found:
\[ AP = \sqrt{a^2 - p^2} - \sqrt{b^2 - p^2} \]

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

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