If the line x `cos theta+y sin theta=P` is the normal to the curve `(x+a)y=1,then theta` may lie in

Options

A. I quadrant
B. II quadrant
C. III quadrant
D. IV quadrant

Answer

To solve the problem step by step, we need to determine the conditions under which the line \( x \cos \theta + y \sin \theta = P \) is the normal to the curve \( (x + a)y = 1 \).

### Step 1: Rewrite the curve equation
The curve is given by:
\[ (x + a)y = 1 \]
We can rewrite this in the form of \( y \):
\[ y = \frac{1}{x + a} \]

### Step 2: Differentiate the curve to find the slope of the tangent
To find the slope of the tangent to the curve, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = -\frac{1}{(x + a)^2} \]

### Step 3: Find the slope of the normal
The slope of the normal is the negative reciprocal of the slope of the tangent:
\[ \text{slope of normal} = -\frac{1}{\frac{dy}{dx}} = -\left(-\frac{(x + a)^2}{1}\right) = (x + a)^2 \]

### Step 4: Find the slope of the normal from the line equation
The equation of the normal line is given by:
\[ x \cos \theta + y \sin \theta = P \]
We can rewrite this in the slope-intercept form \( y = mx + c \):
\[ y \sin \theta = -x \cos \theta + P \]
\[ y = -\frac{\cos \theta}{\sin \theta} x + \frac{P}{\sin \theta} \]
Thus, the slope of the normal from this equation is:
\[ \text{slope of normal} = -\cot \theta \]

### Step 5: Set the slopes equal
Since both expressions represent the slope of the normal, we can equate them:
\[ (x + a)^2 = -\cot \theta \]

### Step 6: Analyze the conditions
Since \( (x + a)^2 \) is always non-negative (i.e., \( \geq 0 \)), we have:
\[ -\cot \theta \geq 0 \implies \cot \theta \leq 0 \]

### Step 7: Determine the quadrants for \( \theta \)
The condition \( \cot \theta \leq 0 \) implies that \( \theta \) must be in the second or fourth quadrant. In these quadrants:
- In the second quadrant, \( \sin \theta > 0 \) and \( \cos \theta < 0 \) (thus \( \cot \theta < 0 \)).
- In the fourth quadrant, \( \sin \theta < 0 \) and \( \cos \theta > 0 \) (thus \( \cot \theta < 0 \)).

### Conclusion
Thus, the values of \( \theta \) may lie in the second and fourth quadrants.