Explanation
Solution
1. Standardize the Parabola Equation
Complete the square for y:
Comparing this to (y−k)2=4a(x−h):
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4a=8⟹a=2
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Vertex (h,k)=(−1,2)
2. Locate the Focus
The focus of (y−k)2=4a(x−h) is (h+a,k):
3. Equation of the Focal Chord
The chord passes through the Focus (1,2) and has an x-intercept of 3, meaning it passes through (3,0).
The slope m of the chord is:
4. Length of the Chord
For a parabola (y−k)2=4a(x−h), the length of a focal chord making an angle θ with the axis of the parabola is 4acsc2θ.
Here, the slope m=−1, so tanθ=−1, which means θ=135∘ or 45∘.