To find the equation of a quadratic graph defined by its focus and directrix, we can utilize the definition that the set of all points on the graph is equidistant from the focus and the directrix.
Given:
- Focus: (6, 0)
- Directrix: y = 10
Let’s denote a point on the parabola as (x, y). The distance from this point to the focus (6, 0) is:
Dfocus = √((x – 6)2 + (y – 0)2)
The distance from the point (x, y) to the directrix (y = 10) is simply the vertical distance:
Ddirectrix = |y – 10|
According to the definition of a parabola, these distances must be equal:
√((x – 6)2 + y2) = |y – 10|
Next, we square both sides to remove the square root, but we need to consider both cases for the absolute value.
1. **Case 1**: When y ≥ 10
(x – 6)2 + y2 = (y – 10)2
Expanding this gives us:
(x – 6)2 + y2 = y2 – 20y + 100
We can simplify by subtracting y2 from both sides:
(x – 6)2 = -20y + 100
Rearranging this gives:
y = -&frac{1}{20}(x – 6)2 + 5
2. **Case 2**: When y < 10
(x – 6)2 + y2 = (10 – y)2
Expanding provides:
(x – 6)2 + y2 = 100 – 20y + y2
Canceling y2:
(x – 6)2 = 100 – 20y
Then rearranging gives:
y = -&frac{1}{20}(x – 6)2 + 5
In both cases, we find that the equation of the parabola is:
y = -&frac{1}{20}(x – 6)2 + 5
This parabola opens downward and has its vertex at the point (6, 5).
Therefore, the equation of the quadratic graph with focus (6, 0) and directrix y = 10 is:
y = -&frac{1}{20}(x – 6)2 + 5