Values of m, for which the line `y=mx+2sqrt5` is a tangent to the hyperbola `16x^(2)-9y^(2)=144`, are the roots of the equation `x^(2)-(a+b)x-4=0`, then the value of `(a+b)` is equal to

Answer

To solve the problem, we need to find the values of \( m \) for which the line \( y = mx + 2\sqrt{5} \) is a tangent to the hyperbola \( 16x^2 - 9y^2 = 144 \). The roots of the quadratic equation formed by these values of \( m \) will help us determine \( a + b \).

### Step 1: Rewrite the hyperbola equation
First, we rewrite the equation of the hyperbola in standard form. We divide the entire equation by 144:

\[ \frac{16x^2}{144} - \frac{9y^2}{144} = 1 \]

This simplifies to:

\[ \frac{x^2}{9} - \frac{y^2}{16} = 1 \]

From this, we identify \( a^2 = 9 \) and \( b^2 = 16 \), giving us \( a = 3 \) and \( b = 4 \).

### Step 2: Use the tangent line condition
The equation of the tangent to the hyperbola can be expressed as:

\[ y = mx \pm \sqrt{a^2 m^2 - b^2} \]

For the line \( y = mx + 2\sqrt{5} \) to be a tangent, we set:

\[ 2\sqrt{5} = \sqrt{a^2 m^2 - b^2} \]

### Step 3: Square both sides
Squaring both sides gives:

\[ (2\sqrt{5})^2 = a^2 m^2 - b^2 \]

This simplifies to:

\[ 20 = 9m^2 - 16 \]

### Step 4: Rearranging the equation
Rearranging the equation leads to:

\[ 9m^2 = 20 + 16 \]
\[ 9m^2 = 36 \]
\[ m^2 = 4 \]

### Step 5: Finding values of m
Taking the square root, we find:

\[ m = 2 \quad \text{or} \quad m = -2 \]

### Step 6: Forming the quadratic equation
The values of \( m \) are the roots of the quadratic equation:

\[ x^2 - (2 + (-2))x - 4 = 0 \]

This simplifies to:

\[ x^2 - 0x - 4 = 0 \]

### Step 7: Identifying a and b
From the quadratic equation \( x^2 - (a + b)x - 4 = 0 \), we see that:

- The sum of the roots \( (a + b) = 0 \)
- The product of the roots is \( -4 \)

### Final Step: Conclusion
Thus, the value of \( a + b \) is:

\[ \boxed{0} \]