Chords of the hyperbola `x^2/a^2-y^2/b^2=1` are tangents to the circle drawn on the line joining the foci asdiameter. Find the locus of the point of intersection of tangents at the extremities of the chords.

Options

A. `(x^2)/(a^4)+(y^2)/(b^4)=(1)/(ab)`
B. `(x^2)/(a^2)+(y^2)/(b^2)=(1)/(a^2-b^2)`
C. `(x^2)/(a^4)+(y^2)/(b^4)=(1)/(a^2+b^2)`
D. `(x^2)/(a^2)+(y^2)/(b^2)=(1)/(b^2-a^2)`

Answer

To find the locus of the point of intersection of tangents at the extremities of the chords of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) that are tangents to the circle drawn on the line joining the foci as diameter, we will follow these steps:

### Step 1: Identify the foci of the hyperbola
The foci of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are located at \( (ae, 0) \) and \( (-ae, 0) \), where \( e = \sqrt{1 + \frac{b^2}{a^2}} \).

### Step 2: Write the equation of the circle
The circle drawn on the line joining the foci as diameter has its center at the origin \( (0, 0) \) and its radius equal to half the distance between the foci. The radius is \( ae \), so the equation of the circle is:
\[ x^2 + y^2 = (ae)^2 \]

### Step 3: Parametrize points on the hyperbola
Let \( P \) and \( Q \) be points on the hyperbola. We can parametrize these points as:
\[ P(a \sec \theta, b \tan \theta) \quad \text{and} \quad Q(a \sec (\theta + \pi), b \tan (\theta + \pi) \]
This gives us:
\[ Q(-a \sec \theta, -b \tan \theta) \]

### Step 4: Find the equation of the chord joining points P and Q
The equation of the chord joining points \( P \) and \( Q \) can be derived using the two-point form of the line:
\[ \frac{x}{a \sec \theta} + \frac{y}{b \tan \theta} = 1 \]

### Step 5: Find the intersection of tangents at P and Q
The tangents at points \( P \) and \( Q \) can be expressed as:
\[ \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1 \]
For point \( P \):
\[ \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1 \quad \Rightarrow \quad \frac{xx}{a^2} - \frac{yy}{b^2} = 1 \]
For point \( Q \):
\[ \frac{xx_2}{a^2} - \frac{yy_2}{b^2} = 1 \]

### Step 6: Find the coordinates of the point of intersection
Let the point of intersection of the tangents be \( (h, k) \). The coordinates can be expressed in terms of \( \theta \):
\[ h = a \frac{\cos \theta}{\cos^2 \theta} \quad \text{and} \quad k = b \frac{\sin \theta}{\cos^2 \theta} \]

### Step 7: Eliminate the parameter to find the locus
To find the locus, we eliminate \( \theta \) from the equations for \( h \) and \( k \):
\[ h = a \sec \theta \quad \text{and} \quad k = b \tan \theta \]
From \( h \), we have \( \sec \theta = \frac{h}{a} \), hence \( \cos \theta = \frac{a}{h} \). Substituting \( \cos \theta \) into the equation for \( k \):
\[ k = b \tan \theta = b \frac{\sin \theta}{\cos \theta} = b \frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta} = b \frac{\sqrt{1 - \left(\frac{a}{h}\right)^2}}{\frac{a}{h}} = \frac{b h}{a} \sqrt{1 - \frac{a^2}{h^2}} \]

### Step 8: Rearranging to find the locus equation
After some algebraic manipulation, we can derive the locus equation:
\[ \frac{h^2}{a^2} + \frac{k^2}{b^2} = 1 \]

This represents another hyperbola.

### Final Answer
Thus, the locus of the point of intersection of the tangents at the extremities of the chords is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]