The number of tangents to `x^(2)//25-y^(2)//9=1` through (5,0) is

Options

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

Answer

To find the number of tangents to the hyperbola given by the equation \( \frac{x^2}{25} - \frac{y^2}{9} = 1 \) through the point \( (5, 0) \), we can follow these steps:

### Step 1: Rewrite the hyperbola equation
We start with the equation of the hyperbola:
\[ \frac{x^2}{25} - \frac{y^2}{9} = 1 \]
We can rearrange this equation to form \( S \):
\[ S = \frac{x^2}{25} - \frac{y^2}{9} - 1 \]

### Step 2: Substitute the point into the equation
Next, we substitute the coordinates of the point \( (5, 0) \) into the expression for \( S \):
\[ S(5, 0) = \frac{5^2}{25} - \frac{0^2}{9} - 1 \]
Calculating this gives:
\[ S(5, 0) = \frac{25}{25} - 0 - 1 = 1 - 1 = 0 \]

### Step 3: Analyze the result
Since \( S(5, 0) = 0 \), this indicates that the point \( (5, 0) \) lies on the hyperbola.

### Step 4: Determine the number of tangents
For any point that lies on the hyperbola, there is exactly one tangent that can be drawn to the hyperbola at that point. Therefore, the number of tangents from the point \( (5, 0) \) to the hyperbola is:
\[ \text{Number of tangents} = 1 \]

### Final Answer
Thus, the number of tangents to the hyperbola through the point \( (5, 0) \) is \( 1 \).

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