Question

The equation of the parabola whose vertex is (-1, -2), axis is vertical and which passes through the point (3, 6), is

• A. \(x^2 + 2x - 2y - 3 = 0\)
• B. \(2x^2 = 3y\)
• C. \(x^2 - 2x + 2y - 3 = 0\)
• D. \(x^2 - 2x - 2y - 3 = 0\)

Hint:

For vertical axis parabola: \((x - h)^2 = 4a(y - k)\), opens upward if \(a>0\).

Answer 1

The Correct Option is A

To find the equation of the parabola with a vertex at \((-1, -2)\), axis vertical, and passing through the point \((3, 6)\), we follow these steps:

1. The standard form of a parabola with a vertical axis and vertex \((h, k)\) is: \((x - h)^2 = 4p(y - k)\) where \(p\) is the distance from the vertex to the focus.
2. For this problem, the vertex is given as \((-1, -2)\). Substituting these into the equation, we get: \((x + 1)^2 = 4p(y + 2)\)
3. We also know that the parabola passes through the point \((3, 6)\). Substituting these coordinates into the equation: \((3 + 1)^2 = 4p(6 + 2)\)
4. Solving for \(p\):
   • \(4^2 = 4p \cdot 8\)
   • \(16 = 32p\)
   • \(p = \frac{1}{2}\)
5. Substituting \(p = \frac{1}{2}\) back into the parabola equation: \((x + 1)^2 = 2(y + 2)\)
6. Expanding and rearranging the equation:
   • Expand the left side: \(x^2 + 2x + 1\)
   • The equation becomes: \(x^2 + 2x + 1 = 2y + 4\)
   • Simplify and rearrange to find the standard form: \(x^2 + 2x - 2y - 3 = 0\)

Therefore, the correct equation of the parabola is \(x^2 + 2x - 2y - 3 = 0\), which matches the given correct answer option: \(\boxed{x^2 + 2x - 2y - 3 = 0}\).

We can verify this by checking each step and ensuring that the calculations correctly lead to this outcome, reinforcing our understanding of parabolas and their properties.