Question

The coordinates of the focus and the vertex of a parabola, respectively, are $(-1,\,4)$ and $(3,\,4)$. Then the equation of the parabola is

• A. $(x-3)^2 = 16(y-4)$
• B. $(x-3)^2 = -16(y-4)$
• C. $(y-4)^2 = -8(x-3)$
• D. $(y-4)^2 = -16(x-3)$
• E. $(y-4)^2 = 1(x-3)$

Hint:

If the focus and vertex have the same $y$-coordinate, the parabola is horizontal. If focus is left of vertex, it opens left: $(y-k)^2 = -4a(x-h)$. If focus is right, it opens right: $(y-k)^2 = 4a(x-h)$.

Answer 1

The Correct Option is D

Step 1: Concept:
• The axis of a parabola passes through both the vertex and the focus.
• The distance between vertex and focus gives the value of \(a\).
• Direction (left/right/up/down) determines the sign in the equation.

Step 2: Identify Key Elements:
• Vertex: \((3,\,4)\)
• Focus: \((-1,\,4)\)
• Since both have same \(y\)-coordinate: \[ \text{Axis is horizontal } (y = 4) \]
• Distance between vertex and focus: \[ |a| = |3 - (-1)| = 4 \]
• Focus lies to the left of vertex: \[ \Rightarrow \text{Parabola opens left} \]

Step 3: Form Equation:
• Standard form for horizontal parabola: \[ (y - k)^2 = -4a(x - h) \]
• Substituting \(a = 4\), \((h,k) = (3,4)\): \[ (y - 4)^2 = -16(x - 3) \]

Step 4: Final Answer:
• \[ (y - 4)^2 = -16(x - 3) \]