Step 1: Understanding the Question:
This question is from coordinate geometry and involves finding the intersection points of a circle and a parabola to determine the length of their common chord.
We need to calculate the length of the common chord ($L_1$), find the latus rectum of the parabola ($L_2$), and then compare their values.
Step 2: Key Formulas and Approach:
Latus Rectum of a Parabola: For a parabola $y^2 = 4ax$, the length of the latus rectum is $4a$.
Common Chord Length: Solve the equations of the circle $x^2 + y^2 = 9$ and the parabola $y^2 = 8x$ simultaneously to find their points of intersection. The distance between these points along the vertical line is the length of the common chord.
Step 3: Detailed Explanation:
Calculate $L_2$:
The equation of the parabola is $y^2 = 8x$.
Comparing this with $y^2 = 4ax$, the length of the latus rectum is:
\[
L_2 = 8 \quad \cdots (1)
\]
Find the Intersection Points:
Substitute the parabola equation $y^2 = 8x$ into the circle equation $x^2 + y^2 = 9$:
\[
x^2 + 8x = 9 \quad \implies \quad x^2 + 8x - 9 = 0
\]
Factoring the quadratic equation:
\[
(x + 9)(x - 1) = 0 \quad \implies \quad x = 1 \text{ or } x = -9
\]
Since $y^2 = 8x$, a negative $x$-value ($x = -9$) would yield imaginary values for $y$. Thus, we choose the valid real solution $x = 1$.
Calculate the Coordinates:
For $x = 1$:
\[
y^2 = 8(1) = 8 \quad \implies \quad y = \pm\sqrt{8} = \pm2\sqrt{2}
\]
The intersection points are $(1, 2\sqrt{2})$ and $(1, -2\sqrt{2})$.
Calculate the Chord Length ($L_1$):
The length of the common chord is the distance between these two points:
\[
L_1 = 2\sqrt{2} - (-2\sqrt{2}) = 4\sqrt{2}\text{ units}
\]
Approximating the value:
\[
L_1 = 4 \times 1.414 = 5.66\text{ units}
\]
Compare $L_1$ and $L_2$:
Comparing our results, we have $L_1 = 5.66$ and $L_2 = 8$.
Since $5.66<8$, we conclude that $L_1<L_2$.
Step 4: Final Answer:
The correct relationship is $L_1<L_2$, which corresponds to Option (C).