If `m_1,m_2` are the slopes of the two tangents that are drawn from (2,3) to the parabola `y^2=4x` , then the value of `1/m_1+1/m_2` is

Options

A. -3
B. 3
C. `2/3`
D. `3/2`

Answer

To solve the problem of finding the value of \( \frac{1}{m_1} + \frac{1}{m_2} \) for the slopes \( m_1 \) and \( m_2 \) of the tangents drawn from the point (2,3) to the parabola \( y^2 = 4x \), we can follow these steps:

### Step 1: Identify the equation of the parabola
The given parabola is \( y^2 = 4x \). This is a standard form of a parabola that opens to the right.

### Step 2: Write the equation of the tangent
For a parabola of the form \( y^2 = 4ax \), the equation of the tangent at a point with slope \( m \) is given by:
\[ y = mx + \frac{a}{m} \]
Here, \( a = 1 \) (since \( 4a = 4 \)), so the equation of the tangent becomes:
\[ y = mx + \frac{1}{m} \]

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

### Step 4: Multiply through by \( m \) to eliminate the fraction
To eliminate the fraction, multiply the entire equation by \( m \):
\[ 3m = 2m^2 + 1 \]

### Step 5: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ 2m^2 - 3m + 1 = 0 \]

### Step 6: Solve the quadratic equation for \( m \)
We can use the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -3 \), and \( c = 1 \):
\[ m = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} \]
\[ m = \frac{3 \pm \sqrt{9 - 8}}{4} \]
\[ m = \frac{3 \pm 1}{4} \]
This gives us two solutions:
\[ m_1 = \frac{4}{4} = 1 \quad \text{and} \quad m_2 = \frac{2}{4} = \frac{1}{2} \]

### Step 7: Calculate \( \frac{1}{m_1} + \frac{1}{m_2} \)
Now, we compute \( \frac{1}{m_1} + \frac{1}{m_2} \):
\[ \frac{1}{m_1} + \frac{1}{m_2} = \frac{1}{1} + \frac{1}{\frac{1}{2}} = 1 + 2 = 3 \]

### Final Answer
Thus, the value of \( \frac{1}{m_1} + \frac{1}{m_2} \) is \( \boxed{3} \).