To find the vertex, focus, equation of directrix, and axis of the parabola given by the equation \(3x^2 - 9x + 5y = 0\), we will follow these steps:
### Step 1: Rearrange the Equation
We start with the equation:
\[
3x^2 - 9x + 5y = 0
\]
Rearranging gives:
\[
5y = -3x^2 + 9x
\]
or
\[
y = -\frac{3}{5}x^2 + \frac{9}{5}x
\]
### Step 2: Complete the Square
Next, we will complete the square for the \(x\) terms:
\[
y = -\frac{3}{5}(x^2 - 3x)
\]
To complete the square inside the parentheses, we take half of the coefficient of \(x\) (which is \(-3\)), square it, and add/subtract it:
\[
y = -\frac{3}{5}\left(x^2 - 3x + \frac{9}{4} - \frac{9}{4}\right)
\]
This simplifies to:
\[
y = -\frac{3}{5}\left((x - \frac{3}{2})^2 - \frac{9}{4}\right)
\]
Distributing the \(-\frac{3}{5}\):
\[
y = -\frac{3}{5}(x - \frac{3}{2})^2 + \frac{27}{20}
\]
### Step 3: Identify Vertex
The vertex form of the parabola is:
\[
y = a(x - h)^2 + k
\]
From our equation, we identify:
- \(h = \frac{3}{2}\)
- \(k = \frac{27}{20}\)
Thus, the vertex is:
\[
\text{Vertex} = \left(\frac{3}{2}, \frac{27}{20}\right)
\]
### Step 4: Find Focus and Directrix
The standard form of a parabola that opens downwards is:
\[
y = -\frac{1}{4b}(x - h)^2 + k
\]
Here, we have:
\[
-\frac{3}{5} = -\frac{1}{4b} \implies 4b = \frac{5}{3} \implies b = \frac{5}{12}
\]
The focus is located at:
\[
\left(h, k - b\right) = \left(\frac{3}{2}, \frac{27}{20} - \frac{5}{12}\right)
\]
To compute \(k - b\), we need a common denominator (60):
\[
\frac{27}{20} = \frac{81}{60}, \quad \frac{5}{12} = \frac{25}{60}
\]
Thus,
\[
k - b = \frac{81}{60} - \frac{25}{60} = \frac{56}{60} = \frac{14}{15}
\]
So, the coordinates of the focus are:
\[
\text{Focus} = \left(\frac{3}{2}, \frac{14}{15}\right)
\]
### Step 5: Equation of the Directrix
The equation of the directrix is given by:
\[
y = k + b = \frac{27}{20} + \frac{5}{12}
\]
Again, using a common denominator (60):
\[
\frac{27}{20} = \frac{81}{60}, \quad \frac{5}{12} = \frac{25}{60}
\]
Thus,
\[
y = \frac{81}{60} + \frac{25}{60} = \frac{106}{60} = \frac{53}{30}
\]
So, the equation of the directrix is:
\[
\text{Directrix: } y = \frac{53}{30}
\]
### Step 6: Axis of the Parabola
The axis of the parabola is the vertical line that passes through the vertex, given by:
\[
x = \frac{3}{2}
\]
### Summary of Results
- **Vertex**: \(\left(\frac{3}{2}, \frac{27}{20}\right)\)
- **Focus**: \(\left(\frac{3}{2}, \frac{14}{15}\right)\)
- **Equation of Directrix**: \(y = \frac{53}{30}\)
- **Axis**: \(x = \frac{3}{2}\)
