Question

The distance of the point \[ (6,4\sqrt{3}) \] from the focus of \[ y^2=8x \] is:

• A. \(64\)
• B. \(4\)
• C. \(8\)
• D. \(2\)

Hint:

For the parabola \[ y^2=4ax, \] the focus is always at \[ (a,0). \] After finding the focus, apply the distance formula directly.

Answer 1

The Correct Option is C

Step 1: Identify the parabola parameters.
Compare $y^2 = 8x$ with the standard form $y^2 = 4ax$. We get $4a = 8$, so $a = 2$.

Step 2: Use the focal distance formula.
For any point $(x_0, y_0)$ on the parabola $y^2 = 4ax$, the distance to the focus equals $x_0 + a$ (the focal distance formula). This is simpler than using the distance formula directly.

Step 3: Verify the point lies on the parabola.
Check $(6, 4\sqrt{3})$: $y^2 = (4\sqrt{3})^2 = 48$ and $8x = 8 \times 6 = 48$. Yes, the point is on the parabola.

Step 4: Apply the focal distance formula.
Distance from $(6, 4\sqrt{3})$ to the focus $= x_0 + a = 6 + 2 = 8$.

Step 5: Confirm using the focus coordinates.
Focus is at $(a, 0) = (2, 0)$. Distance $= \sqrt{(6-2)^2 + (4\sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8$. Both methods agree.

Step 6: State the answer.
The distance from $(6, 4\sqrt{3})$ to the focus is $8$.
\[ \boxed{8} \]