Question

The equation of the parabola with focus at $(3, 0)$ and directrix $x + 3 = 0$ is:

• A. $y^2 = 12x$
• B. $y^2 = -12x$
• C. $x^2 = 12y$
• D. $x^2 = -12y$

Hint:

If the focus is on the positive x-axis $(a, 0)$ and directrix is parallel to the y-axis, the parabola opens rightwards with the standard form $y^2 = 4ax$.

Answer 1

The Correct Option is A

Step 1: Use the focus-directrix setup.
The focus is at $(3,0)$ on the positive x-axis and the directrix is $x = -3$. This is a sideways-opening parabola with vertex midway.

Step 2: Find the vertex.
The vertex is halfway between the focus $(3,0)$ and the foot of the directrix $(-3,0)$, which is $(0,0)$.

Step 3: Pick the standard form.
Vertex at origin opening right gives $y^2 = 4ax$.

Step 4: Find $a$.
The focus is at distance $a$ from the vertex, so $a = 3$.

Step 5: Substitute.
\[ y^2 = 4(3)x \]

Step 6: Final equation.
\[ y^2 = 12x \] \[ \boxed{ y^2 = 12x } \]