Question

find the vertex and focus of the parabola: x² - 10x - 12y - 23 = 0 vertex = (?, ) focus = (, )

Explanation:

Step1: Rewrite the equation in vertex - form

First, complete the square for the \(x\) terms.

$$
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$$

The standard form of a parabola is \((x - h)^{2}=4p(y - k)\), where \((h,k)\) is the vertex.

Step2: Identify the vertex

Comparing \((x - 5)^{2}=12(y + 4)\) with \((x - h)^{2}=4p(y - k)\), we have \(h = 5\) and \(k=-4\). So the vertex \((h,k)=(5,-4)\)

Step3: Find the value of \(p\)

Since \(4p = 12\), then \(p = 3\)

Step4: Calculate the focus

The focus of the parabola \((x - h)^{2}=4p(y - k)\) is \((h,k + p)\)
Substitute \(h = 5\), \(k=-4\) and \(p = 3\)
\(k+p=-4 + 3=-1\)
So the focus is \((5,-1)\)

Answer:

Vertex \(=(5,-4)\)
Focus \(=(5,-1)\)