The minimum area of the triangle formed by any tangent to the ellipse `x^2/16+y^2/81=1` and the coordinate axes is :

Options

A. 12
B. 18
C. 26
D. 36

Answer

To find the minimum area of the triangle formed by any tangent to the ellipse \( \frac{x^2}{16} + \frac{y^2}{81} = 1 \) and the coordinate axes, we can follow these steps:

### Step 1: Identify the ellipse parameters
The given ellipse is in the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a^2 = 16 \) and \( b^2 = 81 \). Therefore, \( a = 4 \) and \( b = 9 \). The major axis is along the y-axis since \( b > a \).

### Step 2: Write the equation of the tangent
The equation of the tangent to the ellipse at a point \( (4 \cos \theta, 9 \sin \theta) \) is given by:
\[ \frac{x \cdot 4 \cos \theta}{16} + \frac{y \cdot 9 \sin \theta}{81} = 1 \]
This simplifies to:
\[ x \cos \theta + y \frac{9}{4} \sin \theta = 1 \]

### Step 3: Find the intercepts on the axes
To find the x-intercept (let \( y = 0 \)):
\[ x \cos \theta = 1 \implies x = \frac{1}{\cos \theta} \]
To find the y-intercept (let \( x = 0 \)):
\[ y \frac{9}{4} \sin \theta = 1 \implies y = \frac{4}{9 \sin \theta} \]

### Step 4: Calculate the area of the triangle
The area \( A \) of the triangle formed by the tangent and the coordinate axes is given by:
\[ A = \frac{1}{2} \times \text{(base)} \times \text{(height)} = \frac{1}{2} \times \left(\frac{1}{\cos \theta}\right) \times \left(\frac{4}{9 \sin \theta}\right) \]
This simplifies to:
\[ A = \frac{2}{9 \cos \theta \sin \theta} = \frac{2}{9} \cdot \frac{2}{\sin(2\theta)} = \frac{4}{9 \sin(2\theta)} \]

### Step 5: Minimize the area
To minimize the area \( A \), we need to maximize \( \sin(2\theta) \). The maximum value of \( \sin(2\theta) \) is 1, which occurs when \( 2\theta = 90^\circ \) or \( \theta = 45^\circ \).

Substituting this back into the area formula:
\[ A_{\text{min}} = \frac{4}{9 \cdot 1} = \frac{4}{9} \]

### Step 6: Final area calculation
Thus, the minimum area of the triangle formed by any tangent to the ellipse and the coordinate axes is:
\[ \text{Minimum Area} = 4 \]