Number of points on the ellipse `(x^(2))/(25) + (y^(2))/(16) =1` from which pair of perpendicular tangents are drawn to the ellipse `(x^(2))/(16) + (y^(2))/(9) =1` is

Options

A. 2
B. 3
C. 4
D. 1

Answer

To solve the problem of finding the number of points on the ellipse \(\frac{x^2}{25} + \frac{y^2}{16} = 1\) from which pairs of perpendicular tangents can be drawn to the ellipse \(\frac{x^2}{16} + \frac{y^2}{9} = 1\), we will follow these steps:

### Step 1: Identify the Director Circle of the Second Ellipse
The equation of the director circle for an ellipse given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is given by:
\[ x^2 + y^2 = a^2 + b^2 \]
For the ellipse \(\frac{x^2}{16} + \frac{y^2}{9} = 1\), we have \(a^2 = 16\) and \(b^2 = 9\). Therefore, the equation of the director circle is:
\[ x^2 + y^2 = 16 + 9 = 25 \]

### Step 2: Set Up the Equations
Now we have two equations:
1. The equation of the director circle:
\[ x^2 + y^2 = 25 \]
2. The equation of the first ellipse:
\[ \frac{x^2}{25} + \frac{y^2}{16} = 1 \]

### Step 3: Substitute for \(y^2\)
From the director circle equation, we can express \(y^2\) in terms of \(x^2\):
\[ y^2 = 25 - x^2 \]

### Step 4: Substitute \(y^2\) into the Ellipse Equation
Substituting \(y^2\) into the ellipse equation gives:
\[ \frac{x^2}{25} + \frac{25 - x^2}{16} = 1 \]

### Step 5: Clear the Denominators
To eliminate the fractions, we can multiply through by the least common multiple of the denominators, which is 400:
\[ 16x^2 + 25(25 - x^2) = 400 \]
This simplifies to:
\[ 16x^2 + 625 - 25x^2 = 400 \]

### Step 6: Combine Like Terms
Combining like terms results in:
\[ -9x^2 + 625 = 400 \]
Rearranging gives:
\[ -9x^2 = 400 - 625 \]
\[ -9x^2 = -225 \]

### Step 7: Solve for \(x^2\)
Dividing both sides by -9:
\[ x^2 = \frac{225}{9} = 25 \]
Thus, we find:
\[ x = \pm 5 \]

### Step 8: Find Corresponding \(y\) Values
Now, substituting \(x = 5\) and \(x = -5\) back into the equation for \(y^2\):
1. For \(x = 5\):
\[ y^2 = 25 - 5^2 = 25 - 25 = 0 \implies y = 0 \]
2. For \(x = -5\):
\[ y^2 = 25 - (-5)^2 = 25 - 25 = 0 \implies y = 0 \]

### Step 9: Identify Points
Thus, the points from which pairs of perpendicular tangents can be drawn are:
\[ (5, 0) \quad \text{and} \quad (-5, 0) \]

### Conclusion
The number of points on the first ellipse from which pairs of perpendicular tangents can be drawn to the second ellipse is:
\[ \boxed{2} \]