The coordinates of the focus of the parabola \(-0.25x^2 = y - 2\) are:
A. (0, 6)
B. (0, 1)
C. (0, −2)
D. (0, 3)
Answer (3)
The equation is − 0.25 x 2 − x + 2 = 0 The coordinates are ( 2 a − b ; 4 a − Δ ) Δ = b 2 − 4 a c = 1 + 2 = 3 The coordinates, thus, are ( − 0.5 1 ; − 1 − 1 ) = ( − 2 ; 1 )
The coordinates of the focus of the given parabola -0.25x² = y - 2 are (0, 1), after rearranging the equation into standard form and calculating the focus using the formula related to the vertex form of a parabola.
To determine the coordinates of the focus of the given parabola -0.25x² = y - 2, we first rewrite the equation in the standard form of a parabola. Starting with the given equation, we add 2 to both sides to get y = -0.25x² + 2. This is in the form of y = a(x - h)² + k, where the vertex (h, k) is at (0, 2), because h = 0 and k = 2 in this equation. Given that the parabola opens downwards (a negative coefficient for x²), the focus is below the vertex.
In the standard form y = a(x - h)² + k, the focus is at (h, k + p), where p = 1/(4a). For our equation, a = -0.25 and the focus is (0, 2 + p). We find p by calculating 1/(4a), thus p = -1.
Therefore, the focus is at (0, 2 - 1), which simplifies to (0, 1).
The coordinates of the focus of the parabola − 0.25 x 2 = y − 2 are (0, 1), found by first rearranging the equation to identify the vertex and then calculating the focus using the formula involving p . Therefore, the correct answer is option B.
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