The locus of a point from which two tangent are drawn to `x^2-y^2=a^2` which are inclined at angle `(pi)/(4)` to each other is

Options

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

Answer

To find the locus of a point from which two tangents are drawn to the hyperbola \(x^2 - y^2 = a^2\) that are inclined at an angle of \(\frac{\pi}{4}\) to each other, we can follow these steps:

### Step 1: Understand the equation of the hyperbola
The given hyperbola is \(x^2 - y^2 = a^2\). The standard form of this hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{a^2} = 1\).

### Step 2: Use the formula for the pair of tangents
The equation of the pair of tangents from a point \((h, k)\) to the hyperbola can be expressed as:
\[ S S_1 = T^2 \]
where \(S = x^2 - y^2 - a^2\), \(S_1 = h^2 - k^2 - a^2\), and \(T = xh - yk\).

### Step 3: Set up the equation
Substituting into the equation, we have:
\[ (x^2 - y^2 - a^2)(h^2 - k^2 - a^2) = (xh - yk)^2 \]

### Step 4: Expand and simplify
Expanding both sides:
1. Left-hand side:
\[ (x^2 - y^2 - a^2)(h^2 - k^2 - a^2) = x^2h^2 - x^2k^2 - x^2a^2 - y^2h^2 + y^2k^2 + y^2a^2 - a^2h^2 + a^2k^2 + a^4 \]
2. Right-hand side:
\[ (xh - yk)^2 = x^2h^2 - 2xyhk + y^2k^2 \]

### Step 5: Rearrange the equation
Now, we equate the two sides:
\[ x^2h^2 - y^2k^2 - x^2a^2 + y^2a^2 - a^2h^2 + a^2k^2 + a^4 = x^2h^2 - 2xyhk + y^2k^2 \]
This simplifies to:
\[ - x^2a^2 + y^2a^2 - a^2h^2 + a^2k^2 + a^4 + 2xyhk = 0 \]

### Step 6: Use the angle condition
The tangents are inclined at an angle of \(\frac{\pi}{4}\), which means:
\[ \tan\left(\frac{\pi}{4}\right) = 1 = \frac{2\sqrt{h^2k^2 - ab}}{a + b} \]
where \(a\) and \(b\) are the coefficients of \(x^2\) and \(y^2\) respectively.

### Step 7: Solve for the locus
From the condition of tangents, we can derive:
\[ h^2 + k^2 = 4a^2 \]
This implies that the locus of the point \((h, k)\) is a circle with radius \(2a\).

### Final Result
Thus, the locus of the point from which two tangents are drawn to the hyperbola \(x^2 - y^2 = a^2\) that are inclined at an angle of \(\frac{\pi}{4}\) to each other is given by:
\[ h^2 + k^2 = 4a^2 \]