If `m_(1) and m_(2)` are slopes of the tangents to the ellipse `(x^(2))/(16)+(y^(2))/(9)=1` which passes through (5, 4), then the value of `(m_(1)+m_(2))-(m_(1)m_(2))` is equal to

Options

A. `(47)/(9)`
B. `-(40)/(6)`
C. `(22)/(3)`
D. `(11)/(3)`

Answer

To solve the problem, we need to find the value of \((m_1 + m_2) - (m_1 m_2)\) where \(m_1\) and \(m_2\) are the slopes of the tangents to the ellipse given by the equation

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

that pass through the point \((5, 4)\).

### Step 1: Identify the parameters of the ellipse
The given ellipse can be rewritten in standard form:
- \(a^2 = 16\) (thus \(a = 4\))
- \(b^2 = 9\) (thus \(b = 3\))

### Step 2: Write the equation of the tangent line
The equation of the tangent to the ellipse at a point \((x_0, y_0)\) can be expressed as:
\[ y = mx \pm \sqrt{a^2 m^2 + b^2} \]
For our ellipse, this becomes:
\[ y = mx \pm \sqrt{16m^2 + 9} \]

### Step 3: Substitute the point (5, 4) into the tangent equation
Since the tangent passes through the point \((5, 4)\), we substitute \(x = 5\) and \(y = 4\):
\[ 4 = 5m \pm \sqrt{16m^2 + 9} \]

### Step 4: Rearrange the equation
This gives us two equations to consider:
1. \(4 = 5m + \sqrt{16m^2 + 9}\)
2. \(4 = 5m - \sqrt{16m^2 + 9}\)

We will focus on the first equation and square both sides to eliminate the square root:
\[ (4 - 5m)^2 = 16m^2 + 9 \]

### Step 5: Expand and simplify
Expanding the left side:
\[ 16 - 40m + 25m^2 = 16m^2 + 9 \]

Rearranging gives:
\[ 25m^2 - 16m^2 - 40m + 16 - 9 = 0 \]
\[ 9m^2 - 40m + 7 = 0 \]

### Step 6: Use the quadratic formula
The roots of this quadratic equation, \(m_1\) and \(m_2\), can be found using the quadratic formula:
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 9\), \(b = -40\), and \(c = 7\).

### Step 7: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \(m_1 + m_2 = -\frac{b}{a} = \frac{40}{9}\)
- The product of the roots \(m_1 m_2 = \frac{c}{a} = \frac{7}{9}\)

### Step 8: Calculate \((m_1 + m_2) - (m_1 m_2)\)
Now we can find the desired expression:
\[ (m_1 + m_2) - (m_1 m_2) = \frac{40}{9} - \frac{7}{9} = \frac{33}{9} = \frac{11}{3} \]

### Final Answer
Thus, the value of \((m_1 + m_2) - (m_1 m_2)\) is:
\[ \frac{11}{3} \]