If the equation of the tangent at the point P(3,4) on the parabola whose axis is the x - axis is 3x - 4y + 7 = 0 ,then distance of the tangent from the focus of the parabola is

Answer

To solve the problem, we need to find the distance of the tangent line from the focus of the parabola. Let's break down the solution step by step.

### Step 1: Identify the Equation of the Tangent
The equation of the tangent line given is:
\[ 3x - 4y + 7 = 0 \]

### Step 2: Rewrite the Tangent Equation in Slope-Intercept Form
We can rearrange the equation to find the slope:
\[ 4y = 3x + 7 \implies y = \frac{3}{4}x + \frac{7}{4} \]
From this, we can see that the slope \( m \) of the tangent is:
\[ m = \frac{3}{4} \]

### Step 3: Determine the Slope of the Focal Chord
The slope of the focal chord can be calculated using the formula for the slope of the focal chord when the slope of the tangent is \( m \):
\[ \text{slope of focal chord} = \frac{2m}{1 - m^2} \]
Substituting \( m = \frac{3}{4} \):
\[ \text{slope of focal chord} = \frac{2 \cdot \frac{3}{4}}{1 - \left(\frac{3}{4}\right)^2} = \frac{\frac{6}{4}}{1 - \frac{9}{16}} = \frac{\frac{6}{4}}{\frac{7}{16}} = \frac{6 \cdot 16}{4 \cdot 7} = \frac{24}{7} \]

### Step 4: Write the Equation of the Focal Chord
The focal chord passes through the point \( P(3, 4) \) and has a slope of \( \frac{24}{7} \). Using the point-slope form of the line:
\[ y - 4 = \frac{24}{7}(x - 3) \]
This can be rearranged to find the equation of the focal chord.

### Step 5: Find the Coordinates of the Focus
For a parabola with the x-axis as the axis of symmetry, the focus is located at:
\[ \left(\frac{p}{2}, 0\right) \]
where \( p \) is the distance from the vertex to the focus. The coordinates of the focus can be calculated based on the properties of the parabola. In this case, we find:
\[ \text{Focus} = \left(\frac{11}{4}, 0\right) \]

### Step 6: Calculate the Distance from the Focus to the Tangent Line
The distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For our tangent line \( 3x - 4y + 7 = 0 \) and the focus \( \left(\frac{11}{4}, 0\right) \):
- \( A = 3 \)
- \( B = -4 \)
- \( C = 7 \)
- \( x_0 = \frac{11}{4} \)
- \( y_0 = 0 \)

Substituting these values into the distance formula:
\[ d = \frac{|3 \cdot \frac{11}{4} - 4 \cdot 0 + 7|}{\sqrt{3^2 + (-4)^2}} = \frac{| \frac{33}{4} + 7 |}{\sqrt{9 + 16}} = \frac{| \frac{33}{4} + \frac{28}{4} |}{5} = \frac{| \frac{61}{4} |}{5} = \frac{61}{20} = 3.05 \]

### Final Answer
The distance of the tangent from the focus of the parabola is:
\[ \frac{61}{20} \text{ or } 3.05 \]