Answer:
Vertex: (1, 5)
Focus: (0, 5)
Step-by-step explanation:
Given that the parabola's equation is y² - 10y + 4x + 21 = 0, solve for the vertex and the focus.
Vertex:
Step 1: Isolate x to the left side of the equation; we get x = - (y² / 4) + (5y / 2) - (21 / 4).
Step 2: Complete the square of the previous equation; here we get - (1 / 4) * (y - 5)² + 1. x is squal to this.
Step 3: Use the vertex form, x = a(y - k)² + 1. a is - 1 / 4, h is 1, and k is 5.
Step 4: Fill in (h, k). We get the answer (1, 5) as the vertex.
Focus:
Step 1: Find p, and distance from the vertex to the focus. We can find p by using 1 / (4a).
Step 2: Simplify. This gives us -1 as p.
Step 3: We plug in the values we just got earlier to get the vertex. The focus of a parabola can be found by using: (h + p, k).
Step 4: Substitude the values. We have (1 + -1, 5).
Step 5: Simplify. Therefore, the answer is (0, 5).
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