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What is the focus of the parabola y=-10x^(2) ? Simplify any fractions. ◻ ◻ 믐

What is the focus of the parabola y=−10x2 y=-10 x^{2} ?\newlineSimplify any fractions.\newline□ \square \newline□ \square \newline믐

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Full solution

Q. What is the focus of the parabola y=−10x2 y=-10 x^{2} ?\newlineSimplify any fractions.\newline□ \square \newline□ \square \newline믐
  1. Standard Form of Parabola: The standard form of a parabola is y=ax2y = ax^2. Here, a=−10a = -10.
  2. Formula for Focus: The focus of a parabola is given by the formula (h,k+14a)(h, k + \frac{1}{4a}), where the vertex is at (h,k)(h, k) and the parabola opens downwards since aa is negative.
  3. Calculate Vertex: For the parabola y=−10x2y = -10x^2, the vertex is at (0,0)(0, 0) because it's in the form y=ax2y = ax^2 without any shifts.
  4. Calculate Y-coordinate: Now, let's calculate the y-coordinate of the focus using k+14ak + \frac{1}{4a}. Here, k=0k = 0 and a=−10a = -10.
  5. Substitute and Solve: Substitute a=−10a = -10 into the formula to get the yy-coordinate of the focus: 0+14(−10)=1−40=−1400 + \frac{1}{4(-10)} = \frac{1}{-40} = -\frac{1}{40}.
  6. Final Focus Coordinates: So, the focus of the parabola y=−10x2y = -10x^2 is at (0,−140)(0, -\frac{1}{40}).

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