Question

Find the equation of the parabola with focus at \( (3,1) \) and vertex at \( (5,1) \).

• A. \( (y - 1)^2 = -8(x - 5) \)
• B. \( (y - 1)^2 = 8(x - 5) \)
• C. \( (y - 1)^2 = 8(x - 3) \)
• D. \( (y - 1)^2 = -8(x - 3) \)
• E. \( (y - 1)^2 = -4(x - 5) \)

Hint:

For a parabola \( (y - k)^2 = 4p(x - h) \), \( p \) is the directed distance from the vertex to the focus.

Answer 1

The Correct Option is A

The equation of a horizontal parabola is: \[ (y - k)^2 = 4p(x - h) \] where \( (h, k) \) is the vertex, and \( p \) is the distance from the vertex to the focus. Given: \[ (h, k) = (5,1), \quad (f_x, f_y) = (3,1) \] \[ p = f_x - h = 3 - 5 = -2 \] Substituting into the equation: \[ (y - 1)^2 = 4(-2)(x - 5) \] \[ (y - 1)^2 = -8(x - 5) \] Thus, the correct answer is: 
Final Answer: \[ \boxed{(y - 1)^2 = -8(x - 5)} \]