Consider a parabola P touches coordinate axes at (4,0) and (0,3).
Length of latus rectum of parabola P is

Answer

To solve the problem, we need to find the length of the latus rectum of the parabola that touches the coordinate axes at the points (4, 0) and (0, 3).

### Step-by-Step Solution:

1. **Identify the Properties of the Parabola:**
The parabola touches the x-axis at (4, 0) and the y-axis at (0, 3). This means that the vertex of the parabola is at the point where it is tangent to the axes.

2. **Determine the Vertex:**
The vertex of the parabola can be considered as the point (h, k) where:
- h = 4 (x-coordinate of the point where it touches the x-axis)
- k = 3 (y-coordinate of the point where it touches the y-axis)

Therefore, the vertex is at (4, 3).

3. **Equation of the Parabola:**
Since the parabola opens downwards (touching the x-axis) and has its vertex at (4, 3), we can express its equation in the form:
\[ (y - k) = a(x - h)^2 \]
Substituting the vertex coordinates:
\[ (y - 3) = a(x - 4)^2 \]

4. **Finding the Value of 'a':**
To find 'a', we use the fact that the parabola touches the x-axis at (4, 0). Substituting this point into the equation:
\[ 0 - 3 = a(4 - 4)^2 \]
This simplifies to:
\[ -3 = 0 \quad \text{(which does not help)} \]
Instead, we can consider the distance from the vertex to the focus (OF) and the directrix. The distance OF is equal to \( \frac{1}{2} \) the length of the latus rectum.

5. **Using the Relationship of Focus and Directrix:**
The distance from the vertex to the focus (OF) is given by:
\[ OF = \sqrt{(4 - 0)^2 + (3 - 0)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

6. **Length of the Latus Rectum:**
The length of the latus rectum (L) is given by the formula:
\[ L = 4p \]
where \( p \) is the distance from the vertex to the focus. Since we found that \( OF = 5 \), we can say that:
\[ L = 4 \times 5 = 20 \]

### Final Answer:
The length of the latus rectum of the parabola P is **20**.