Find the vertex, axis, focus, and latus rectum of the parabola 4y²+12x–20y+67=0

The equation can be written y² – 5y = –3x – 67/4

Step 1: Complete the square in y

(y – 5/2)² = –3x – 67/4 + 25/4

(y – 5/2)² = –3x – 42/4

(y – 5/2)² = –3(x + 7/2)

Step 2: Transform coordinates

Let new origin be (–7/2, 5/2). Then equation becomes: 

Y² = –3X

Step 3: Properties of parabola

• Standard form: Y² = –4aX ⇒ 4a = 3 ⇒ a = 3/4
• Axis: parallel to X-axis (horizontal)
• Concavity: opens to negative X-axis
• Latus rectum = 3

Step 4: Referred to original axes

• Vertex: (–7/2, 5/2)
• Axis: y = 5/2
• Focus: (–7/2 – 3/4, 5/2) = (–17/4, 5/2)