Find the focus of the parabola defined by the equation left bracket, x, minus, 2, right bracket, squared, equals, minus, 32, left bracket, y, plus, 6, right bracket(x−2) 2 =−32(y+6).
Question
Find the focus of the parabola defined by the equation left bracket, x, minus, 2, right bracket, squared, equals, minus, 32, left bracket, y, plus, 6, right bracket(x−2) 2 =−32(y+6).
Solution
The equation of the parabola is given as (x - 2)^2 = -32(y + 6).
This is in the form of (x - h)^2 = 4a(y - k), where (h, k) is the vertex of the parabola and 'a' is the distance from the vertex to the focus.
From the equation, we can see that h = 2, k = -6, and 4a = -32.
Solving for 'a', we get a = -32 / 4 = -8.
The focus of a parabola in this form is given by the point (h, k - a).
Substituting the values we found for h, k, and a, the focus is (2, -6 - (-8)) = (2, 2).
So, the focus of the parabola is at the point (2, 2).
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