A tangent to the ellipse `x^2+4y^2=4` meets the ellipse `x^2+2y^2=6` at P&Q. The angle between the tangents at P and Q of the ellipse x 2 +2y 2 =6 is

Options

A. `pi//2`
B. `pi//3`
C. `pi//4`
D. `pi//6`

Answer

To solve the problem, we need to find the angle between the tangents at points P and Q on the ellipse defined by the equation \(x^2 + 2y^2 = 6\), where the tangent intersects the ellipse defined by \(x^2 + 4y^2 = 4\).

### Step-by-Step Solution:

1. **Identify the Ellipses**:
The first ellipse is given by:
\[ \frac{x^2}{4} + \frac{y^2}{1} = 1 \]
This means \(a^2 = 4\) and \(b^2 = 1\), hence \(a = 2\) and \(b = 1\).

The second ellipse is given by:
\[ \frac{x^2}{6} + \frac{y^2}{3} = 1 \]
This means \(a^2 = 6\) and \(b^2 = 3\), hence \(a = \sqrt{6}\) and \(b = \sqrt{3}\).

2. **Find the Foci of the Second Ellipse**:
The foci of the ellipse \(x^2/a^2 + y^2/b^2 = 1\) are located at \((\pm c, 0)\), where \(c = \sqrt{a^2 - b^2}\).
For the second ellipse:
\[ c = \sqrt{6 - 3} = \sqrt{3} \]
Thus, the foci are at \((\pm \sqrt{3}, 0)\).

3. **Equation of the Tangent to the First Ellipse**:
The equation of the tangent to the ellipse \(x^2 + 4y^2 = 4\) at a point \((x_0, y_0)\) is given by:
\[ \frac{xx_0}{4} + \frac{yy_0}{1} = 1 \]
This tangent will intersect the second ellipse at points P and Q.

4. **Finding the Points of Intersection**:
Substitute the tangent equation into the second ellipse's equation:
\[ \frac{x^2}{6} + \frac{y^2}{3} = 1 \]
We can express \(y\) from the tangent equation and substitute it into this equation to find the coordinates of points P and Q.

5. **Finding the Angle Between the Tangents**:
The angle \(\theta\) between the tangents at points P and Q can be calculated using the formula:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
where \(m_1\) and \(m_2\) are the slopes of the tangents at points P and Q.

To find the slopes, differentiate the equation of the second ellipse implicitly or use the coordinates of points P and Q.

6. **Final Calculation**:
After finding the slopes \(m_1\) and \(m_2\), substitute them into the angle formula to find the angle \(\theta\).