Angle between the tangents drawn from point (4, 5) to the ellipse `x^2/16 +y^2/25 =1` is

Options

A. `pi/3`
B. `5pi/6`
C. `pi/4`
D. `pi/2`

Answer

To find the angle between the tangents drawn from the point (4, 5) 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 ellipse equation and the point
The equation of the ellipse is:
\[ \frac{x^2}{16} + \frac{y^2}{25} = 1 \]
The point from which tangents are drawn is \(P(4, 5)\).

### Step 2: Write the combined equation of the tangents
The combined equation of the tangents from point \(P\) to the ellipse can be expressed as:
\[ S S_1 = T^2 \]
where:
- \(S\) is the equation of the ellipse,
- \(S_1\) is obtained by substituting the coordinates of point \(P\) into the ellipse equation,
- \(T\) is the equation of the tangent line.

### Step 3: Calculate \(S\)
The equation of the ellipse can be rewritten as:
\[ S: \frac{x^2}{16} + \frac{y^2}{25} - 1 = 0 \]

### Step 4: Calculate \(S_1\)
Substituting the point \(P(4, 5)\) into the ellipse equation:
\[ S_1 = \frac{4^2}{16} + \frac{5^2}{25} - 1 = \frac{16}{16} + \frac{25}{25} - 1 = 1 + 1 - 1 = 1 \]

### Step 5: Write the equation for \(T\)
The equation for \(T\) is given by:
\[ T: \frac{x}{4} + \frac{y}{5} - 1 = 0 \]

### Step 6: Combine the equations
Now we have:
\[ S S_1 = T^2 \]
Substituting \(S\) and \(S_1\):
\[ \left(\frac{x^2}{16} + \frac{y^2}{25} - 1\right)(1) = \left(\frac{x}{4} + \frac{y}{5} - 1\right)^2 \]

### Step 7: Expand and rearrange
Expanding the right-hand side:
\[ \left(\frac{x}{4} + \frac{y}{5} - 1\right)^2 = \frac{x^2}{16} + \frac{y^2}{25} + 1 - 2\left(\frac{x}{4} + \frac{y}{5}\right) \]
Setting this equal to the left-hand side gives us the combined equation of the tangents.

### Step 8: Identify coefficients for angle calculation
The general form of the equation of the pair of straight lines is:
\[ Ax^2 + By^2 + 2Hxy = 0 \]
From our equation, we can identify:
- \(A = 0\) (since there is no \(x^2\) term),
- \(B = 0\) (since there is no \(y^2\) term),
- \(H\) can be derived from the expanded form.

### Step 9: Calculate the angle between the tangents
The formula for the angle \(\theta\) between the two tangents is given by:
\[ \tan \theta = \frac{2\sqrt{H^2 - AB}}{A + B} \]
Since \(A = 0\) and \(B = 0\), we find that:
\[ A + B = 0 \implies \tan \theta \text{ is undefined} \]
This indicates that the angle \(\theta\) is \(90^\circ\) or \(\frac{\pi}{2}\).

### Final Answer
Thus, the angle between the tangents drawn from the point (4, 5) to the ellipse is:
\[ \theta = \frac{\pi}{2} \]