Question

For the parabola \(4(y - 1)^2 = -7(x - 3)\), the coordinates of vertex, focus and length of the latus rectum are respectively:

• A. (1,3); (41/16, 1); 7/2
• B. (3,1); (41/16, 1); 7/4
• C. (3,1); (41/16, 0); 7/4
• D. (3,1); (0, 41/16); 7/2

Hint:

Exam Tip:
For parabolas:

• \((y - k)^2 = 4a(x - h)\) opens right, focus \((h + a, k)\).
• \((y - k)^2 = -4a(x - h)\) opens left, focus \((h - a, k)\).
• \((x - h)^2 = 4a(y - k)\) opens up, focus \((h, k + a)\).
• \((x - h)^2 = -4a(y - k)\) opens down, focus \((h, k - a)\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to identify the vertex, focus, and length of the latus rectum of a parabola given in standard form.

Step 2: Key Formula or Approach:
The given equation is \(4(y - 1)^2 = -7(x - 3)\).
This is a parabola of the form \((y - k)^2 = -4a(x - h)\), where the vertex is \((h, k)\), focus is \((h - a, k)\), and the length of the latus rectum is \(4a\).

Step 3: Detailed Explanation:
Rewrite the equation: \[ (y - 1)^2 = -\frac{7}{4}(x - 3) \] Compare with \((y - k)^2 = -4a(x - h)\): \[ k = 1, \quad h = 3, \quad 4a = \frac{7}{4} \Rightarrow a = \frac{7}{16} \] Vertex: \((h, k) = (3, 1)\).
Focus: Since the parabola opens to the left (negative sign), the focus is \((h - a, k) = \left(3 - \frac{7}{16}, 1\right) = \left(\frac{48 - 7}{16}, 1\right) = \left(\frac{41}{16}, 1\right)\).
Length of latus rectum: \(4a = \frac{7}{4}\).

Step 4: Final Answer:
Vertex = (3, 1), Focus = (41/16, 1), Length of latus rectum = 7/4.
This matches option (B). Therefore, option (B) is correct.