The number of tangents to `(x^(2))/(25)+(y^(2))/(9)=1` through (1,1) is

Options

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

Answer

To find the number of tangents to the ellipse given by the equation \(\frac{x^2}{25} + \frac{y^2}{9} = 1\) through the point (1, 1), we can follow these steps:

### Step 1: Identify the parameters of the ellipse
The standard form of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Here, \(a^2 = 25\) and \(b^2 = 9\). Therefore, \(a = 5\) and \(b = 3\).

### Step 2: Set up the equation for tangents
We will use the condition for the number of tangents from a point \((h, k)\) to the ellipse. The condition is based on the value of \(S_1\), where:
\[ S = \frac{x^2}{25} + \frac{y^2}{9} - 1 \]
Substituting the point (1, 1) into this equation gives:
\[ S_1 = \frac{1^2}{25} + \frac{1^2}{9} - 1 \]

### Step 3: Calculate \(S_1\)
Now, we compute \(S_1\):
\[ S_1 = \frac{1}{25} + \frac{1}{9} - 1 \]
To combine the fractions, we find a common denominator. The least common multiple of 25 and 9 is 225. Thus:
\[ S_1 = \frac{9}{225} + \frac{25}{225} - \frac{225}{225} \]
\[ S_1 = \frac{9 + 25 - 225}{225} = \frac{34 - 225}{225} = \frac{-191}{225} \]

### Step 4: Analyze the value of \(S_1\)
Since \(S_1 < 0\), we can apply the rule for the number of tangents:
- If \(S_1 < 0\), there are 0 tangents.
- If \(S_1 = 0\), there is 1 tangent.
- If \(S_1 > 0\), there are 2 tangents.

### Conclusion
Since \(S_1 < 0\), the number of tangents to the ellipse from the point (1, 1) is:
\[ \text{Number of tangents} = 0 \]

### Final Answer
The number of tangents to the ellipse through the point (1, 1) is **0**.

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