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

### Step 1: Rewrite the equation of the ellipse in standard form
The equation of the ellipse is given as:
\[ 3x^2 + 5y^2 = 32 \]
To rewrite it in standard form, we divide the entire equation by 32:
\[ \frac{x^2}{\frac{32}{3}} + \frac{y^2}{\frac{32}{5}} = 1 \]
This gives us:
\[ \frac{x^2}{\frac{32}{3}} + \frac{y^2}{\frac{32}{5}} = 1 \]
where \(a^2 = \frac{32}{3}\) and \(b^2 = \frac{32}{5}\).

### Step 2: Substitute the point into the ellipse equation
Next, we need to check the position of the point \((3, 5)\) with respect to the ellipse. We substitute \(x = 3\) and \(y = 5\) into the left-hand side of the ellipse equation:
\[ 3(3^2) + 5(5^2) = 3(9) + 5(25) = 27 + 125 = 152 \]

### Step 3: Compare with the right-hand side
Now we compare this value with the right-hand side of the ellipse equation, which is 32:
\[ 152 > 32 \]
Since the value we calculated (152) is greater than 32, this indicates that the point \((3, 5)\) lies outside the ellipse.

### Step 4: Determine the number of tangents
From the properties of conic sections, we know that if a point lies outside an ellipse, we can draw exactly two tangents from that point to the ellipse. Therefore, the number of real tangents that can be drawn from the point \((3, 5)\) to the ellipse \(3x^2 + 5y^2 = 32\) is:
\[ \text{Number of tangents} = 2 \]

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

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