Consider a parabola P touches coordinate axes at (4,0) and (0,3).
if focus of parabola P is (a,b) then the value of b-a is

Options

A. `1/25`
B. `3/25`
C. `4/25`
D. `12/25`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
The parabola touches the coordinate axes at points (4,0) and (0,3). This means that these points are tangents to the parabola, and the parabola opens towards the origin.

### Step 2: Determine the equations for slopes
Since the parabola touches the axes, we can find the slopes of the lines from the focus (a,b) to the points of tangency (4,0) and (0,3).

1. The slope of line Pb (from focus (a,b) to point (0,3)):
\[ \text{slope of } Pb = \frac{b - 3}{a - 0} = \frac{b - 3}{a} \]

2. The slope of line Pa (from focus (a,b) to point (4,0)):
\[ \text{slope of } Pa = \frac{b - 0}{a - 4} = \frac{b}{a - 4} \]

### Step 3: Set up the perpendicularity condition
Since these two lines are perpendicular, the product of their slopes must equal -1:
\[ \left(\frac{b - 3}{a}\right) \cdot \left(\frac{b}{a - 4}\right) = -1 \]
Cross-multiplying gives:
\[ (b - 3)b = -a(a - 4) \]
This simplifies to:
\[ b^2 - 3b + a^2 - 4a = 0 \quad \text{(Equation 1)} \]

### Step 4: Set up the second perpendicularity condition
For the second pair of slopes (the slopes from the origin (0,0) to the points of tangency), we have:
1. Slope of line OP (from origin to (4,0)):
\[ \text{slope of } OP = \frac{0 - b}{0 - a} = \frac{-b}{-a} = \frac{b}{a} \]

2. Slope of line OQ (from origin to (0,3)):
\[ \text{slope of } OQ = \frac{3 - 0}{0 - 0} \text{ (undefined)} \]

Since OP and OQ are also perpendicular:
\[ \frac{b}{a} \cdot \frac{3}{0} \text{ (undefined)} \]
This condition is satisfied since we are considering the slopes of the tangents.

### Step 5: Use the relationship between a and b
From the relationship between the slopes derived earlier, we can also express:
\[ 4a = 3b \quad \text{(Equation 2)} \]

### Step 6: Solve the equations
Now we have two equations:
1. \( b^2 - 3b + a^2 - 4a = 0 \)
2. \( 4a - 3b = 0 \)

From Equation 2, we can express \( a \) in terms of \( b \):
\[ a = \frac{3b}{4} \]

Substituting this into Equation 1:
\[ b^2 - 3b + \left(\frac{3b}{4}\right)^2 - 4\left(\frac{3b}{4}\right) = 0 \]
This simplifies to:
\[ b^2 - 3b + \frac{9b^2}{16} - 3b = 0 \]
Combining like terms gives:
\[ \frac{25b^2}{16} - 6b = 0 \]
Factoring out \( b \):
\[ b\left(\frac{25b}{16} - 6\right) = 0 \]
Thus, \( b = 0 \) or \( b = \frac{96}{25} \).

### Step 7: Find a using b
Using \( b = \frac{48}{25} \):
\[ a = \frac{3 \cdot \frac{48}{25}}{4} = \frac{36}{25} \]

### Step 8: Calculate \( b - a \)
Now, we can find \( b - a \):
\[ b - a = \frac{48}{25} - \frac{36}{25} = \frac{12}{25} \]

### Final Answer
Thus, the value of \( b - a \) is:
\[ \boxed{\frac{12}{25}} \]