Question

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

• A. y²=3x-9
• B. y²=4x-12
• C. y²=12x
• D. y²=12x-36
• E. y²=12x-9

Answer 1

The Correct Option is C

Step 1: Find the vertex of the parabola  
The vertex of a parabola is the midpoint between the focus and the directrix. 
Given focus: \( (3,0) \) 
Given directrix: \( x + 3 = 0 \Rightarrow x = -3 \) 
The vertex is: \[ \left( \frac{3 + (-3)}{2}, \frac{0 + 0}{2} \right) = \left( \frac{0}{2}, 0 \right) = (0,0) \]

Step 2: Find the equation of the parabola 
The standard equation of a parabola with vertex \( (h,k) \) and focus \( (h + a, k) \) (since it opens rightward) is: \[ (y - k)^2 = 4a (x - h) \] Here, \( h = 0 \), \( k = 0 \), and \( a = \) distance from vertex to focus: \[ a = 3 - 0 = 3 \] So, the equation becomes: \[ y^2 = 4(3)(x - 0) \] \[ y^2 = 12x \]

Final Answer: The equation of the parabola is \( y^2 = 12x \).