Number of perpendicular tangents that can be drawn on the ellipse `(x^(2))/(16)+(y^(2))/(25)=1` from point (6, 7) is

Answer

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

### Step 1: Identify the equation of the ellipse
The given equation of the ellipse is:
\[ \frac{x^2}{16} + \frac{y^2}{25} = 1 \]
From this, we can identify \(a^2 = 16\) and \(b^2 = 25\).

### Step 2: Calculate the semi-major and semi-minor axes
The semi-major axis \(a\) and semi-minor axis \(b\) can be calculated as follows:
\[ a = \sqrt{16} = 4, \quad b = \sqrt{25} = 5 \]

### Step 3: Write the equation of the director circle
The equation of the director circle for an ellipse is given by:
\[ x^2 + y^2 = a^2 + b^2 \]
Calculating \(a^2 + b^2\):
\[ a^2 + b^2 = 16 + 25 = 41 \]
Thus, the equation of the director circle is:
\[ x^2 + y^2 = 41 \]

### Step 4: Check if the point (6, 7) lies on the director circle
Now we need to check if the point (6, 7) satisfies the equation of the director circle:
\[ 6^2 + 7^2 = 36 + 49 = 85 \]
Since \(85 \neq 41\), the point (6, 7) does not lie on the director circle.

### Step 5: Conclusion about the number of perpendicular tangents
Since the point (6, 7) does not lie on the director circle, it implies that there are no perpendicular tangents that can be drawn from this point to the ellipse. Therefore, the number of perpendicular tangents is:
\[ \text{Number of perpendicular tangents} = 0 \]

### Final Answer
The number of perpendicular tangents that can be drawn from the point (6, 7) to the ellipse is \(0\).

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