The angle between the tangents drawn form the point (3, 4) to the parabola `y^(2)-2y+4x=0`, is

Options

A. `tan^(-1)(8sqrt5//7)`
B. `tan^(-1)(12//sqrt5)`
C. `tan^(-1)(sqrt5//7)`
D. none of these

Answer

To find the angle between the tangents drawn from the point (3, 4) to the parabola given by the equation \(y^2 - 2y + 4x = 0\), we will follow these steps:

### Step 1: Convert the parabola to standard form
We start with the equation of the parabola:
\[ y^2 - 2y + 4x = 0 \]
Rearranging gives:
\[ y^2 - 2y = -4x \]
Now, we complete the square for the left side:
\[ (y - 1)^2 - 1 = -4x \]
Adding 1 to both sides:
\[ (y - 1)^2 = -4x + 1 \]
This can be rewritten as:
\[ (y - 1)^2 = -4(x - \frac{1}{4}) \]
This shows that the parabola opens to the left with vertex at \((\frac{1}{4}, 1)\).

### Step 2: Find the equation of the tangents from the point (3, 4)
The general equation of the tangent to the parabola \(y^2 = 4ax\) in slope form is:
\[ y = mx - \frac{a}{m} \]
For our parabola, \(a = 1\) (since \(4a = 4\)). Thus, the tangent equation becomes:
\[ y = mx - \frac{1}{m} \]
We substitute \(y\) with \(y - 1\) and \(x\) with \(x - \frac{1}{4}\):
\[ y - 1 = m\left(x - \frac{1}{4}\right) - \frac{1}{m} \]
Substituting the point (3, 4) into this equation:
\[ 4 - 1 = m\left(3 - \frac{1}{4}\right) - \frac{1}{m} \]
This simplifies to:
\[ 3 = m\left(\frac{11}{4}\right) - \frac{1}{m} \]
Multiplying through by \(m\) to eliminate the fraction:
\[ 3m = \frac{11m^2}{4} - 1 \]
Rearranging gives:
\[ \frac{11m^2}{4} - 3m - 1 = 0 \]
Multiplying through by 4 to clear the fraction:
\[ 11m^2 - 12m - 4 = 0 \]

### Step 3: Use the quadratic formula to find slopes \(m_1\) and \(m_2\)
Using the quadratic formula \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 11\), \(b = -12\), and \(c = -4\):
\[ D = b^2 - 4ac = (-12)^2 - 4 \cdot 11 \cdot (-4) = 144 + 176 = 320 \]
Thus, the slopes are:
\[ m_{1,2} = \frac{12 \pm \sqrt{320}}{22} \]
Calculating \(\sqrt{320} = 8\sqrt{5}\):
\[ m_{1,2} = \frac{12 \pm 8\sqrt{5}}{22} = \frac{6 \pm 4\sqrt{5}}{11} \]

### Step 4: Find the angle between the tangents
The angle \(\theta\) between the two tangents can be found using:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]
Where \(m_1 m_2 = \frac{c}{a} = \frac{-4}{11}\) and \(m_1 - m_2 = \sqrt{D}/a = \frac{8\sqrt{5}}{11}\):
\[ \tan \theta = \frac{\frac{8\sqrt{5}}{11}}{1 - \frac{4}{11}} = \frac{\frac{8\sqrt{5}}{11}}{\frac{7}{11}} = \frac{8\sqrt{5}}{7} \]

### Final Answer
Thus, the angle between the tangents drawn from the point (3, 4) to the parabola is:
\[ \theta = \tan^{-1}\left(\frac{8\sqrt{5}}{7}\right) \]