The chord of contact of a point `P` w.r.t a hyperbola and its auxiliary circle are at right angle. Then the point `P` lies (a) on conjugate hyperbola (b) one of the directrix (c) one of the asymptotes (d) none of these

Options

A. conjugate hyperbola
B. one of the directrix
C. asymptotes
D. none of these

Explanation

To solve the problem step by step, we will analyze the given conditions and derive the necessary equations to determine the location of point \( P \).

### Step 1: Understand the Equations of the Hyperbola and the Auxiliary Circle
The standard equation of a hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
The auxiliary circle of the hyperbola has the equation:
\[ x^2 + y^2 = a^2 \]

### Step 2: Write the Chord of Contact for the Hyperbola
For a point \( P(h, k) \), the chord of contact with respect to the hyperbola is given by:
\[ \frac{hx}{a^2} - \frac{ky}{b^2} = 1 \]

### Step 3: Write the Chord of Contact for the Auxiliary Circle
The chord of contact with respect to the auxiliary circle is given by:
\[ \frac{hx}{a^2} + \frac{ky}{a^2} = 1 \]

### Step 4: Determine the Slopes of the Chords
From the equations of the chords, we can express them in slope-intercept form to find their slopes.

1. For the hyperbola:
\[ ky = \frac{b^2}{a^2}hx - b^2 \implies y = \frac{b^2}{a^2 k}hx - \frac{b^2}{k} \]
The slope \( m_1 = \frac{b^2}{a^2 k}h \).

2. For the auxiliary circle:
\[ ky = -\frac{h}{a^2}x + a^2 \implies y = -\frac{h}{a^2 k}x + \frac{a^2}{k} \]
The slope \( m_2 = -\frac{h}{a^2 k} \).

### Step 5: Use the Condition of Perpendicularity
Since the chords are at right angles, the product of their slopes must equal \(-1\):
\[ m_1 \cdot m_2 = -1 \]
Substituting the slopes:
\[ \left(\frac{b^2}{a^2 k}h\right) \left(-\frac{h}{a^2 k}\right) = -1 \]
This simplifies to:
\[ -\frac{b^2 h^2}{a^4 k^2} = -1 \implies b^2 h^2 = a^4 k^2 \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ \frac{h^2}{a^2} - \frac{k^2}{b^2} = 0 \]
This implies:
\[ \frac{h^2}{a^2} = \frac{k^2}{b^2} \]

### Step 7: Identify the Locus of Point \( P \)
The equation \( \frac{h^2}{a^2} - \frac{k^2}{b^2} = 0 \) can be rewritten as:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \]
This represents the asymptotes of the hyperbola.

### Conclusion
Thus, the point \( P \) lies on one of the asymptotes of the hyperbola.

### Final Answer
The correct option is (c) one of the asymptotes.