Question:
The equation of the parabola with focus at $(3, 0)$ and directrix $x + 3 = 0$ is:
The equation of the parabola with focus at $(3, 0)$ and directrix $x + 3 = 0$ is:
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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 $y^2 = 4ax$.
Updated On: Jun 3, 2026
- $y^2 = 12x$
- $y^2 = -12x$
- $x^2 = 12y$
- $x^2 = -12y$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The standard equation of a parabola with vertex at the origin $(0, 0)$, focus at $(a, 0)$, and directrix $x = -a$ is given by $y^2 = 4ax$.
Step 2: Meaning
Here, the focus is at $(3, 0)$, which lies on the positive x-axis, implying $a = 3$. The directrix is $x = -3 \implies x + 3 = 0$.
Step 3: Analysis
The vertex is the midpoint of the segment joining the focus $(3, 0)$ and the point $(-3, 0)$ on the directrix, which is indeed $(0, 0)$. Using the standard parabola equation for $a = 3$: \[ y^2 = 4(3)x \implies y^2 = 12x \]
Step 4: Conclusion
The equation of the parabola is $y^2 = 12x$.
Final Answer: (A)
The standard equation of a parabola with vertex at the origin $(0, 0)$, focus at $(a, 0)$, and directrix $x = -a$ is given by $y^2 = 4ax$.
Step 2: Meaning
Here, the focus is at $(3, 0)$, which lies on the positive x-axis, implying $a = 3$. The directrix is $x = -3 \implies x + 3 = 0$.
Step 3: Analysis
The vertex is the midpoint of the segment joining the focus $(3, 0)$ and the point $(-3, 0)$ on the directrix, which is indeed $(0, 0)$. Using the standard parabola equation for $a = 3$: \[ y^2 = 4(3)x \implies y^2 = 12x \]
Step 4: Conclusion
The equation of the parabola is $y^2 = 12x$.
Final Answer: (A)
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