To solve the problem, we need to analyze both statements one by one.
### Step 1: Analyze Statement 1
**Statement 1:** The length of the common chord of the parabola \(y^2 = 8x\) and the circle \(x^2 + y^2 = 9\) is less than the length of the latus rectum of the parabola.
1. **Find the length of the common chord:**
- The equation of the parabola is \(y^2 = 8x\).
- The equation of the circle is \(x^2 + y^2 = 9\).
- Substitute \(y^2 = 8x\) into the circle's equation:
\[
x^2 + 8x = 9
\]
- Rearranging gives:
\[
x^2 + 8x - 9 = 0
\]
- To find the roots, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
- Here, \(a = 1\), \(b = 8\), and \(c = -9\).
- The discriminant \(D = b^2 - 4ac = 8^2 - 4 \cdot 1 \cdot (-9) = 64 + 36 = 100\).
- So, the roots are:
\[
x = \frac{-8 \pm 10}{2} = 1 \text{ and } -9
\]
- Since we are interested in the positive x-coordinate, we take \(x = 1\).
2. **Find the corresponding y-coordinates:**
- Substitute \(x = 1\) back into the parabola's equation:
\[
y^2 = 8 \cdot 1 = 8 \implies y = \pm 2\sqrt{2}
\]
- The points of intersection (common chord) are \((1, 2\sqrt{2})\) and \((1, -2\sqrt{2})\).
3. **Calculate the length of the common chord:**
- The length of the common chord is the distance between these two points:
\[
\text{Length} = |y_2 - y_1| = |2\sqrt{2} - (-2\sqrt{2})| = 4\sqrt{2}
\]
4. **Find the length of the latus rectum of the parabola:**
- The formula for the length of the latus rectum of a parabola \(y^2 = 4ax\) is \(4a\).
- Here, \(4a = 8\) implies \(a = 2\).
- Therefore, the length of the latus rectum is \(8\).
5. **Compare the lengths:**
- Length of the common chord: \(4\sqrt{2} \approx 5.66\)
- Length of the latus rectum: \(8\)
- Since \(4\sqrt{2} < 8\), Statement 1 is **True**.
### Step 2: Analyze Statement 2
**Statement 2:** If the vertex of a parabola lies at the point \((a, 0)\) and the directrix is \(x + a = 0\), then the focus of the parabola is at the point \((2a, 0)\).
1. **Understand the properties of the parabola:**
- The vertex is at \((a, 0)\) and the directrix is \(x + a = 0\) or \(x = -a\).
- The focus of a parabola is located at a distance \(p\) from the vertex along the axis of symmetry.
2. **Determine the position of the focus:**
- The distance from the vertex \((a, 0)\) to the directrix \(x = -a\) is:
\[
\text{Distance} = a - (-a) = 2a
\]
- The focus will be located \(p\) units to the right of the vertex, where \(p = 2a\).
- Therefore, the focus is at:
\[
(a + 2a, 0) = (3a, 0)
\]
3. **Conclusion for Statement 2:**
- The statement claims the focus is at \((2a, 0)\), which is incorrect.
- Therefore, Statement 2 is **False**.
### Final Conclusion
- **Statement 1:** True
- **Statement 2:** False
