The angle between tangents drawn to the curve `xy = 4` from the point `(1, 3)` is

Options

A. `30^(@)`
B. `60^(@)`
C. `tan^(-1) . 4/5`
D. `tan^(-1) . 5/4`

Answer

To find the angle between the tangents drawn to the curve \(xy = 4\) from the point \((1, 3)\), we can follow these steps:

### Step 1: Find the equation of the tangents from the point to the curve
The given curve is \(xy = 4\). We can express this in terms of \(y\):
\[ y = \frac{4}{x} \]

### Step 2: Find the slope of the tangents
To find the slope of the tangents, we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = -\frac{4}{x^2} \]

### Step 3: Use the point-slope form of the tangent line
The equation of the tangent line at a point \((t, \frac{4}{t})\) on the curve can be written as:
\[ y - \frac{4}{t} = -\frac{4}{t^2}(x - t) \]
This simplifies to:
\[ y = -\frac{4}{t^2}x + \left(\frac{4}{t} + \frac{4}{t}\right) \]

### Step 4: Substitute the point (1, 3) into the tangent equation
Since the tangents pass through the point \((1, 3)\), we substitute \(x = 1\) and \(y = 3\):
\[ 3 = -\frac{4}{t^2}(1 - t) + \frac{8}{t} \]

### Step 5: Rearrange and simplify the equation
Rearranging gives:
\[ 3 = -\frac{4}{t^2} + \frac{4}{t} + \frac{8}{t} \]
Combining the terms:
\[ 3 = -\frac{4}{t^2} + \frac{12}{t} \]
Multiplying through by \(t^2\) to eliminate the fraction:
\[ 3t^2 = -4 + 12t \]
Rearranging leads to:
\[ 3t^2 - 12t + 4 = 0 \]

### Step 6: Solve for \(t\) using the quadratic formula
Using the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ t = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 3 \cdot 4}}{2 \cdot 3} \]
Calculating the discriminant:
\[ t = \frac{12 \pm \sqrt{144 - 48}}{6} = \frac{12 \pm \sqrt{96}}{6} = \frac{12 \pm 4\sqrt{6}}{6} = 2 \pm \frac{2\sqrt{6}}{3} \]

### Step 7: Find the slopes of the tangents
The slopes \(m_1\) and \(m_2\) corresponding to the values of \(t\) are:
\[ m_1 = -\frac{4}{(2 + \frac{2\sqrt{6}}{3})^2}, \quad m_2 = -\frac{4}{(2 - \frac{2\sqrt{6}}{3})^2} \]

### Step 8: Calculate the angle between the tangents
The angle \(\theta\) between the two tangents can be found using the formula:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

### Step 9: Substitute the slopes into the formula
After calculating \(m_1\) and \(m_2\), substitute them into the formula to find \(\tan \theta\).

### Step 10: Calculate \(\theta\)
Finally, use the inverse tangent function to find \(\theta\):
\[ \theta = \tan^{-1}(\tan \theta) \]