If tangents drawn from the point `(a ,2)` to the hyperbola `(x^2)/(16)-(y^2)/9=1` are perpendicular, then the value of `a^2` is _____

Explanation

To solve the problem step by step, we will follow the reasoning outlined in the video transcript:

### Step 1: Write the equation of the hyperbola
The given hyperbola is:
\[ \frac{x^2}{16} - \frac{y^2}{9} = 1 \]
Here, \(a^2 = 16\) and \(b^2 = 9\).

### Step 2: Write the equation of the tangent
The equation of the tangent to the hyperbola at a point is given by:
\[ y = mx \pm \sqrt{a^2 m^2 - b^2} \]
Substituting \(a^2\) and \(b^2\):
\[ y = mx \pm \sqrt{16m^2 - 9} \]

### Step 3: Substitute the point (a, 2) into the tangent equation
Since the tangents pass through the point \((a, 2)\), we substitute \(x = a\) and \(y = 2\):
\[ 2 = ma \pm \sqrt{16m^2 - 9} \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ 2 - ma = \pm \sqrt{16m^2 - 9} \]

### Step 5: Square both sides
Squaring both sides results in:
\[ (2 - ma)^2 = 16m^2 - 9 \]
Expanding the left side:
\[ 4 - 4ma + m^2a^2 = 16m^2 - 9 \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ m^2a^2 - 16m^2 + 4ma + 13 = 0 \]
This is a quadratic equation in \(m\).

### Step 7: Identify coefficients
The quadratic equation can be expressed as:
\[ m^2(a^2 - 16) + 4am + 13 = 0 \]
Let \(A = a^2 - 16\), \(B = 4a\), and \(C = 13\).

### Step 8: Use the condition for perpendicular tangents
For the tangents to be perpendicular, the product of the slopes \(m_1\) and \(m_2\) (roots of the quadratic) must equal \(-1\):
\[ m_1 m_2 = \frac{C}{A} = -1 \]
Thus,
\[ \frac{13}{a^2 - 16} = -1 \]

### Step 9: Solve for \(a^2\)
Cross-multiplying gives:
\[ 13 = - (a^2 - 16) \]
This simplifies to:
\[ 13 = -a^2 + 16 \]
Rearranging gives:
\[ a^2 = 16 - 13 = 3 \]

### Final Answer
The value of \(a^2\) is:
\[ \boxed{3} \]