If the chords of contact of tangents from two points `(x_(1),y_(1)) and (x_(2),y_(2))` to the elipse`x^(2)/a^(2)+y^(2)/b^(2)=1` are at right angles, then find `(x_(1)x_(2))/(y_(1)y_(2))`

Answer

To solve the problem, we need to find the value of \(\frac{x_1 x_2}{y_1 y_2}\) given that the chords of contact from the points \((x_1, y_1)\) and \((x_2, y_2)\) to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) are at right angles.

### Step-by-Step Solution:

1. **Write the equations of the chords of contact**:
The equation of the chord of contact from the point \((x_1, y_1)\) to the ellipse is given by:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \tag{1} \]
Similarly, the equation of the chord of contact from the point \((x_2, y_2)\) is:
\[ \frac{xx_2}{a^2} + \frac{yy_2}{b^2} = 1 \tag{2} \]

2. **Identify the slopes of the lines**:
From equation (1), the slope \(m_1\) can be determined as follows:
\[ m_1 = -\frac{\text{coefficient of } x}{\text{coefficient of } y} = -\frac{x_1/a^2}{y_1/b^2} = -\frac{b^2 x_1}{a^2 y_1} \]
From equation (2), the slope \(m_2\) is:
\[ m_2 = -\frac{x_2/a^2}{y_2/b^2} = -\frac{b^2 x_2}{a^2 y_2} \]

3. **Use the condition that the lines are perpendicular**:
Since the chords of contact are at right angles, we have:
\[ m_1 \cdot m_2 = -1 \]
Substituting the values of \(m_1\) and \(m_2\):
\[ \left(-\frac{b^2 x_1}{a^2 y_1}\right) \left(-\frac{b^2 x_2}{a^2 y_2}\right) = -1 \]
This simplifies to:
\[ \frac{b^4 x_1 x_2}{a^4 y_1 y_2} = -1 \]

4. **Rearranging the equation**:
Rearranging gives:
\[ \frac{x_1 x_2}{y_1 y_2} = -\frac{a^4}{b^4} \]

5. **Final result**:
Therefore, the required value is:
\[ \frac{x_1 x_2}{y_1 y_2} = -\frac{a^4}{b^4} \]