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 determine the values of \( m \) for which the line \( y = mx + 2\sqrt{5} \) is a tangent to the hyperbola given by \( 16x^2 - 9y^2 = 144 \). We will then relate these values of \( m \) to the equation \( x^2 - (a+b)x - 4 = 0 \) and find \( a + b \).

### Step 1: Rewrite the hyperbola in standard form
The hyperbola equation is given as:
\[ 16x^2 - 9y^2 = 144 \]
Dividing the entire equation by 144, we get:
\[ \frac{x^2}{9} - \frac{y^2}{16} = 1 \]
This shows that \( a^2 = 9 \) and \( b^2 = 16 \).

### Step 2: Use the tangent line equation
The equation of the tangent line to the hyperbola can be expressed as:
\[ y = mx \pm \sqrt{a^2 m^2 - b^2} \]
For our line \( y = mx + 2\sqrt{5} \), we can equate the constant term:
\[ 2\sqrt{5} = \sqrt{9m^2 - 16} \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (2\sqrt{5})^2 = 9m^2 - 16 \]
This simplifies to:
\[ 20 = 9m^2 - 16 \]

### Step 4: Rearranging the equation
Rearranging the equation, we have:
\[ 9m^2 = 20 + 16 \]
\[ 9m^2 = 36 \]
Dividing by 9:
\[ m^2 = 4 \]

### Step 5: Finding values of \( m \)
Taking the square root gives us:
\[ m = \pm 2 \]

### Step 6: Relate \( m \) to the quadratic equation
The values of \( m \) are the roots of the equation \( x^2 - (a+b)x - 4 = 0 \). The roots we found are \( 2 \) and \( -2 \).

### Step 7: Calculate \( a + b \)
Using Vieta's formulas, the sum of the roots \( 2 + (-2) = 0 \) gives us:
\[ -(a + b) = 0 \implies a + b = 0 \]

### Final Answer
Thus, the value of \( a + b \) is:
\[ \boxed{0} \]