If the chords of contact of tangents from two points `(alpha,beta)` and `(gamma, delta)` to the ellipse `x^(2)/5+y^(2)/2=1` are perpendicular, then `(alphagamma)/(betadelta)` =

Options

A. `4/25`
B. `-4/25`
C. `25/4`
D. `-25/4`

Answer

To solve the problem, we need to find the value of \(\frac{\alpha \gamma}{\beta \delta}\) given that the chords of contact of tangents from the points \((\alpha, \beta)\) and \((\gamma, \delta)\) to the ellipse \(\frac{x^2}{5} + \frac{y^2}{2} = 1\) are perpendicular.

### Step 1: Write the equation of the chord of contact
For a point \((x_1, y_1)\), the equation of the chord of contact to the ellipse \(\frac{x^2}{5} + \frac{y^2}{2} = 1\) is given by:
\[ \frac{xx_1}{5} + \frac{yy_1}{2} = 1 \]

### Step 2: Apply the formula for the points \((\alpha, \beta)\) and \((\gamma, \delta)\)
1. For the point \((\alpha, \beta)\):
\[ \frac{x\alpha}{5} + \frac{y\beta}{2} = 1 \quad \text{(1)} \]
2. For the point \((\gamma, \delta)\):
\[ \frac{x\gamma}{5} + \frac{y\delta}{2} = 1 \quad \text{(2)} \]

### Step 3: Find the slopes of the tangents
From equation (1), rearranging gives:
\[ y = -\frac{2\alpha}{5\beta}x + \frac{2}{\beta} \]
Thus, the slope \(m_1\) of the tangent from point \((\alpha, \beta)\) is:
\[ m_1 = -\frac{2\alpha}{5\beta} \]

From equation (2), rearranging gives:
\[ y = -\frac{2\gamma}{5\delta}x + \frac{2}{\delta} \]
Thus, the slope \(m_2\) of the tangent from point \((\gamma, \delta)\) is:
\[ m_2 = -\frac{2\gamma}{5\delta} \]

### Step 4: Use the condition of perpendicularity
Since the tangents are perpendicular, we have:
\[ m_1 \cdot m_2 = -1 \]
Substituting the values of \(m_1\) and \(m_2\):
\[ \left(-\frac{2\alpha}{5\beta}\right) \left(-\frac{2\gamma}{5\delta}\right) = -1 \]
This simplifies to:
\[ \frac{4\alpha\gamma}{25\beta\delta} = -1 \]

### Step 5: Rearranging the equation
Multiplying both sides by \(-25\beta\delta\):
\[ 4\alpha\gamma = -25\beta\delta \]
Dividing both sides by \(4\beta\delta\):
\[ \frac{\alpha\gamma}{\beta\delta} = -\frac{25}{4} \]

### Final Answer
Thus, the value of \(\frac{\alpha \gamma}{\beta \delta}\) is:
\[ \frac{\alpha \gamma}{\beta \delta} = -\frac{25}{4} \]