Question

The focus of the parabola \(y^2 + 4y - 8x + 20 = 0\) is at the point:

• A. \( (0, -2) \)
• B. \( (2, -2) \)
• C. \( (4, -2) \)
• D. \( (2, 0) \)
• E. \( (4, -4) \)

Hint:

When completing the square for a parabola, make sure to balance the equation by adding and subtracting the same values. Remember, the focus of a parabola \((y-k)^2 = 4p(x-h)\) lies \(p\) units from the vertex along the axis of symmetry.

Answer 1

The Correct Option is C

First, rewrite the equation \(y^2 + 4y - 8x + 20 = 0\) in a form that reveals the vertex and direction: \[ y^2 + 4y = 8x - 20 \] Complete the square for the \(y\)-terms: \[ (y+2)^2 - 4 = 8x - 20 \] \[ (y+2)^2 = 8x - 16 \] \[ (y+2)^2 = 8(x-2) \] This is a parabola that opens rightwards with the vertex form \((y-k)^2 = 4p(x-h)\), where \(k = -2\), \(h = 2\), and \(4p = 8\) so \(p = 2\).
The focus of a parabola \( (y-k)^2 = 4p(x-h) \) is at \( (h+p, k) \): \[ (h+p, k) = (2+2, -2) = (4, -2) \]