Find an equation of the tangent to the ellipse `x^2/81+y^2/49=1` at the point P whose eccentric angle is `pi//6`. Also find the coordinates of P .

Answer

To find the equation of the tangent to the ellipse \( \frac{x^2}{81} + \frac{y^2}{49} = 1 \) at the point \( P \) whose eccentric angle is \( \frac{\pi}{6} \), we will follow these steps:

### Step 1: Identify the parameters of the ellipse
The given ellipse is in the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a^2 = 81 \) and \( b^2 = 49 \). Thus, we have:
- \( a = \sqrt{81} = 9 \)
- \( b = \sqrt{49} = 7 \)

### Step 2: Find the coordinates of point \( P \)
The coordinates of a point \( P \) on the ellipse in terms of the eccentric angle \( \theta \) are given by:
\[ x = a \cos(\theta) \quad \text{and} \quad y = b \sin(\theta) \]
Substituting \( a = 9 \), \( b = 7 \), and \( \theta = \frac{\pi}{6} \):
\[ x = 9 \cos\left(\frac{\pi}{6}\right) = 9 \cdot \frac{\sqrt{3}}{2} = \frac{9\sqrt{3}}{2} \]
\[ y = 7 \sin\left(\frac{\pi}{6}\right) = 7 \cdot \frac{1}{2} = \frac{7}{2} \]
Thus, the coordinates of point \( P \) are:
\[ P\left(\frac{9\sqrt{3}}{2}, \frac{7}{2}\right) \]

### Step 3: Find the equation of the tangent at point \( P \)
The equation of the tangent to the ellipse at point \( (x_1, y_1) \) is given by:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
Substituting \( x_1 = \frac{9\sqrt{3}}{2} \), \( y_1 = \frac{7}{2} \), \( a^2 = 81 \), and \( b^2 = 49 \):
\[ \frac{x \cdot \frac{9\sqrt{3}}{2}}{81} + \frac{y \cdot \frac{7}{2}}{49} = 1 \]
Simplifying this:
\[ \frac{9\sqrt{3}x}{162} + \frac{7y}{98} = 1 \]
Multiplying through by the least common multiple of the denominators (which is 162):
\[ 9\sqrt{3}x + \frac{162 \cdot 7y}{98} = 162 \]
Calculating \( \frac{162 \cdot 7}{98} \):
\[ \frac{162 \cdot 7}{98} = \frac{1134}{98} = \frac{567}{49} \]
Thus, the equation becomes:
\[ 9\sqrt{3}x + \frac{567}{49}y = 162 \]

### Final Equation of the Tangent
The final equation of the tangent to the ellipse at point \( P \) is:
\[ 9\sqrt{3}x + \frac{567}{49}y = 162 \]