The locus of the point of intersection of tangents at the extremities of the chords of hyperbola `x^(2)/a^(2) - y^(2)/b^(2) = 1` which are tangents to the circle drawn on the line joining the foci as diameter is

Options

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

Answer

To solve the problem, we need to find the locus of the point of intersection of tangents at the extremities of the chords of the hyperbola given by the equation:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

which are tangents to the circle drawn on the line joining the foci as diameter.

### 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 \((c, 0)\) and \((-c, 0)\), where \(c = \sqrt{a^2 + b^2}\).

### Step 2: Determine the equation of the circle
The circle has its diameter along the line joining the foci. The center of the circle is at the origin \((0, 0)\) and the radius is \(\frac{c}{2} = \frac{\sqrt{a^2 + b^2}}{2}\). Therefore, the equation of the circle is:

\[ x^2 + y^2 = \left(\frac{\sqrt{a^2 + b^2}}{2}\right)^2 = \frac{a^2 + b^2}{4} \]

### Step 3: Find the coordinates of points on the hyperbola
Let \(P\) and \(Q\) be the points on the hyperbola corresponding to angles \(\alpha\) and \(\beta\). The coordinates of these points are:

\[ P = \left(a \sec \alpha, b \tan \alpha\right) \]
\[ Q = \left(a \sec \beta, b \tan \beta\right) \]

### Step 4: Write the equations of the tangents at points \(P\) and \(Q\)
The equations of the tangents at points \(P\) and \(Q\) are given by:

For point \(P\):
\[ \frac{x}{a \sec \alpha} - \frac{y}{b \tan \alpha} = 1 \]

For point \(Q\):
\[ \frac{x}{a \sec \beta} - \frac{y}{b \tan \beta} = 1 \]

### Step 5: Find the intersection of the tangents
To find the intersection of these two tangents, we can solve the two equations simultaneously. This will give us the coordinates \((x, y)\) of the intersection point.

### Step 6: Substitute the intersection point into the circle's equation
The distance from the center of the circle to the line formed by the tangents must equal the radius of the circle. We can use the formula for the distance from a point to a line to set up an equation.

### Step 7: Simplify and find the locus
After substituting the intersection point \((x, y)\) into the equation derived from the circle's radius condition, we will simplify the resulting equation to find the locus of the intersection points.

### Final Result
After performing the necessary algebraic manipulations, we will arrive at the locus equation:

\[ \frac{x^2}{a^4} + \frac{y^2}{b^4} = 1 \]

This indicates that the locus of the point of intersection of the tangents at the extremities of the chords is an ellipse.