The number of real tangents that can be drawn to the ellipse `3x^(2)+5y^(2)=32` passing through (3,5) is

Options

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

Explanation

To determine the number of real tangents that can be drawn to the ellipse \(3x^2 + 5y^2 = 32\) from the point \(P(3, 5)\), we can follow these steps:

### Step 1: Write the equation of the ellipse
The given equation of the ellipse is:
\[ 3x^2 + 5y^2 = 32 \]
We can rewrite this as:
\[ 3x^2 + 5y^2 - 32 = 0 \]
This will help us evaluate the position of point \(P\) relative to the ellipse.

### Step 2: Substitute the coordinates of point \(P\)
Now, we substitute the coordinates of point \(P(3, 5)\) into the equation:
\[ S_1 = 3(3^2) + 5(5^2) - 32 \]
Calculating \(S_1\):
\[ S_1 = 3(9) + 5(25) - 32 = 27 + 125 - 32 = 120 \]

### Step 3: Determine the position of point \(P\)
Now we analyze the value of \(S_1\):
- If \(S_1 = 0\), then point \(P\) lies on the ellipse.
- If \(S_1 < 0\), then point \(P\) lies inside the ellipse.
- If \(S_1 > 0\), then point \(P\) lies outside the ellipse.

Since \(S_1 = 120 > 0\), point \(P(3, 5)\) lies outside the ellipse.

### Step 4: Determine the number of tangents
From the properties of tangents to an ellipse:
- If the point lies on the ellipse, there is 1 tangent.
- If the point lies inside the ellipse, there are 0 tangents.
- If the point lies outside the ellipse, there are 2 tangents.

Since point \(P(3, 5)\) lies outside the ellipse, we conclude that there are 2 tangents that can be drawn from point \(P\) to the ellipse.

### Final Answer
The number of real tangents that can be drawn to the ellipse from the point \(P(3, 5)\) is:
\[ \boxed{2} \]