Number of tangents from `(7,6)` to ellipse `(x^(2))/(16)+(y^(2))/(25)=1` is

Options

A. 0
B. 1
C. 2
D. None of these

Answer

To find the number of tangents from the point (7, 6) to the ellipse given by the equation \(\frac{x^2}{16} + \frac{y^2}{25} = 1\), we can follow these steps:

### Step 1: Write the equation of the ellipse in standard form
The equation of the ellipse is already given in standard form:
\[ \frac{x^2}{16} + \frac{y^2}{25} = 1 \]
This indicates that the semi-major axis \(a = 5\) (along the y-axis) and the semi-minor axis \(b = 4\) (along the x-axis).

### Step 2: Define the function \(P\)
We define the function \(P\) as:
\[ P(x, y) = \frac{x^2}{16} + \frac{y^2}{25} - 1 \]
We will evaluate this function at the point (7, 6).

### Step 3: Substitute the point (7, 6) into \(P\)
Now, we substitute \(x = 7\) and \(y = 6\) into the function \(P\):
\[ P(7, 6) = \frac{7^2}{16} + \frac{6^2}{25} - 1 \]
Calculating each term:
\[ P(7, 6) = \frac{49}{16} + \frac{36}{25} - 1 \]

### Step 4: Find a common denominator and simplify
To combine the fractions, we need a common denominator. The least common multiple of 16 and 25 is 400.
\[ P(7, 6) = \frac{49 \times 25}{400} + \frac{36 \times 16}{400} - 1 \]
Calculating the numerators:
\[ P(7, 6) = \frac{1225}{400} + \frac{576}{400} - 1 \]
Combining the fractions:
\[ P(7, 6) = \frac{1225 + 576}{400} - 1 = \frac{1801}{400} - 1 \]
Converting 1 to a fraction with the same denominator:
\[ P(7, 6) = \frac{1801}{400} - \frac{400}{400} = \frac{1801 - 400}{400} = \frac{1401}{400} \]

### Step 5: Analyze the value of \(P(7, 6)\)
Since \(P(7, 6) = \frac{1401}{400} > 0\), this means that the point (7, 6) lies outside the ellipse.

### Step 6: Determine the number of tangents
According to the properties of tangents to conic sections:
- If the point lies outside the ellipse, there are 2 tangents.
- If the point lies inside the ellipse, there are 0 tangents.
- If the point lies on the ellipse, there is 1 tangent.

Since the point (7, 6) lies outside the ellipse, the number of tangents from (7, 6) to the ellipse is:
\[ \text{Number of tangents} = 2 \]

### Final Answer
The number of tangents from the point (7, 6) to the ellipse is **2**.

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