The number of tangents to `(x^(2))/(9)+(y^(2))/(4)=1` through (3,2) is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the number of tangents to the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) through the point (3, 2), we can follow these steps:

### Step 1: Write the equation of the ellipse in standard form
The given equation of the ellipse is already in standard form:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]

### Step 2: Identify the coordinates of the point
The point through which we need to draw tangents is \( (h, k) = (3, 2) \).

### Step 3: Substitute the point into the ellipse equation
We need to evaluate the expression:
\[ S = \frac{h^2}{9} + \frac{k^2}{4} - 1 \]
Substituting \(h = 3\) and \(k = 2\):
\[ S = \frac{3^2}{9} + \frac{2^2}{4} - 1 \]
Calculating each term:
\[ S = \frac{9}{9} + \frac{4}{4} - 1 = 1 + 1 - 1 \]
Thus,
\[ S = 1 \]

### Step 4: Determine the number of tangents based on the value of S
Now we analyze the value of \(S\):
- If \(S < 0\), there are 0 tangents (the point is inside the ellipse).
- If \(S = 0\), there is 1 tangent (the point is on the ellipse).
- If \(S > 0\), there are 2 tangents (the point is outside the ellipse).

Since \(S = 1\) (which is greater than 0), we conclude that there are 2 tangents from the point (3, 2) to the ellipse.

### Final Answer
The number of tangents to the ellipse through the point (3, 2) is **2**.

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