Questions: Find the vertex and focus of the parabola: x^2+14x-8y+65=0 Vertex = ([?],[]) Focus = ([],[])

Solution Steps

To find the vertex and focus of the parabola given by the equation \(x^2 + 14x - 8y + 65 = 0\), we first need to rewrite the equation in the standard form of a parabola. This involves completing the square for the \(x\) terms. Once in standard form, we can identify the vertex and use the formula for the focus of a parabola that opens vertically.

The given equation of the parabola is:

\[ x^2 + 14x - 8y + 65 = 0 \]

To find the vertex and focus, we need to rewrite this equation in the standard form of a parabola. First, let's isolate the \(y\)-term:

\[ x^2 + 14x + 65 = 8y \]

To complete the square for the \(x\)-terms, we take the coefficient of \(x\), which is 14, divide it by 2, and square it:

\[ \left(\frac{14}{2}\right)^2 = 49 \]

Add and subtract 49 inside the equation:

\[ x^2 + 14x + 49 - 49 + 65 = 8y \]

This simplifies to:

\[ (x + 7)^2 + 16 = 8y \]

Rearrange to isolate \(y\):

\[ (x + 7)^2 = 8y - 16 \]

\[ (x + 7)^2 = 8(y - 2) \]

The standard form of a parabola that opens vertically is \((x - h)^2 = 4p(y - k)\), where \((h, k)\) is the vertex. Comparing this with our equation \((x + 7)^2 = 8(y - 2)\), we identify:

• \(h = -7\)
• \(k = 2\)

Thus, the vertex is \((-7, 2)\).

The value of \(4p\) is 8, so \(p = \frac{8}{4} = 2\). Since the parabola opens upwards, the focus is located \(p\) units above the vertex. Therefore, the focus is:

\[ (-7, 2 + 2) = (-7, 4) \]

Final Answer