If the tangents from the point `(lambda, 3)` to the ellipse `x^2/9+y^2/4=1` are at right angles then `lambda` is

Options

A. `pm1`
B. `pm2`
C. `pm3`
D. `pm4`

Answer

To solve the problem, we need to find the value of \( \lambda \) such that the tangents from the point \( (\lambda, 3) \) to the ellipse \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) are at right angles to each other.

### Step 1: Write the equation of the ellipse
The given ellipse is:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]
Here, \( a^2 = 9 \) (so \( a = 3 \)) and \( b^2 = 4 \) (so \( b = 2 \)).

### Step 2: Use the condition for tangents
The general equation of the tangent to the ellipse at the point \( (x_0, y_0) \) is given by:
\[ \frac{xx_0}{9} + \frac{yy_0}{4} = 1 \]
For the point \( (\lambda, 3) \), we can substitute \( x_0 = \lambda \) and \( y_0 = 3 \):
\[ \frac{x \lambda}{9} + \frac{y \cdot 3}{4} = 1 \]

### Step 3: Rearranging the tangent equation
Rearranging the equation gives:
\[ y = -\frac{4\lambda}{9}x + \frac{12}{4} = -\frac{4\lambda}{9}x + 3 \]
This is in the form \( y = mx + c \) where \( m = -\frac{4\lambda}{9} \) and \( c = 3 \).

### Step 4: Condition for tangents to be perpendicular
For the tangents to be perpendicular, the product of their slopes must equal \(-1\). If \( m_1 \) and \( m_2 \) are the slopes of the tangents, then:
\[ m_1 \cdot m_2 = -1 \]
Using the condition for tangents to the ellipse, we have:
\[ c^2 = a^2 m^2 + b^2 \]
Substituting \( c = 3 - m\lambda \) into the equation gives:
\[ (3 - m\lambda)^2 = 9m^2 + 4 \]

### Step 5: Substitute and simplify
Substituting \( c \) into the equation:
\[ (3 - m\lambda)^2 = 9m^2 + 4 \]
Expanding gives:
\[ 9 - 6m\lambda + m^2\lambda^2 = 9m^2 + 4 \]
Rearranging gives:
\[ m^2\lambda^2 - 15m^2 - 6m\lambda + 5 = 0 \]

### Step 6: Use the quadratic formula
This is a quadratic equation in \( m \). The product of the roots \( m_1 \cdot m_2 = \frac{c}{a} = \frac{5}{\lambda^2 - 9} \). Setting this equal to \(-1\):
\[ \frac{5}{\lambda^2 - 9} = -1 \]
Cross-multiplying gives:
\[ 5 = -(\lambda^2 - 9) \]
Thus,
\[ 5 = -\lambda^2 + 9 \]
Rearranging gives:
\[ \lambda^2 = 4 \]

### Step 7: Solve for \( \lambda \)
Taking the square root gives:
\[ \lambda = \pm 2 \]

### Final Answer
Thus, the values of \( \lambda \) are:
\[ \lambda = 2 \quad \text{or} \quad \lambda = -2 \]