How many real tangents can be drawn to the ellipse `5x^(2)+9y^(2)=32` from the point (2,3)?

Options

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

Explanation

To determine how many real tangents can be drawn to the ellipse \(5x^2 + 9y^2 = 32\) from the point (2, 3), we will follow these steps:

### Step 1: Rewrite the ellipse in standard form
The given equation of the ellipse is \(5x^2 + 9y^2 = 32\). We can rewrite this in standard form by dividing the entire equation by 32:

\[ \frac{x^2}{\frac{32}{5}} + \frac{y^2}{\frac{32}{9}} = 1 \]

This gives us:

\[ \frac{x^2}{\frac{32}{5}} + \frac{y^2}{\frac{32}{9}} = 1 \]

### Step 2: Identify the semi-major and semi-minor axes
From the standard form, we can identify:

- \(a^2 = \frac{32}{5}\) (semi-major axis)
- \(b^2 = \frac{32}{9}\) (semi-minor axis)

### Step 3: Determine the position of the point (2, 3) with respect to the ellipse
To determine the position of the point (2, 3) relative to the ellipse, we substitute \(x = 2\) and \(y = 3\) into the left-hand side of the ellipse equation:

\[ 5(2^2) + 9(3^2) = 5(4) + 9(9) = 20 + 81 = 101 \]

### Step 4: Compare with the right-hand side of the equation
Now, we compare this result with the right-hand side of the ellipse equation:

\[ 101 \quad \text{(from the point)} \quad \text{and} \quad 32 \quad \text{(from the ellipse)} \]

Since \(101 > 32\), we conclude that the point (2, 3) lies outside the ellipse.

### Step 5: Conclusion about the number of tangents
According to the properties of tangents to an ellipse:

- If a point lies outside the ellipse, two tangents can be drawn from that point to the ellipse.
- If a point lies on the ellipse, one tangent can be drawn.
- If a point lies inside the ellipse, no tangents can be drawn.

Since the point (2, 3) lies outside the ellipse, we can draw **two tangents** from this point to the ellipse.

### Final Answer
Thus, the number of real tangents that can be drawn to the ellipse \(5x^2 + 9y^2 = 32\) from the point (2, 3) is **2**.

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