Questions: Find the focus and directrix of the following parabola: (y-1)^2=4(x-1) Focus: ([?], ) Directrix: x=

Find the focus and directrix of the following parabola: (y-1)^2=4(x-1) Focus: ([?], ) Directrix: x=
Transcript text: Find the focus and directrix of the following parabola: \[ (y-1)^{2}=4(x-1) \] Focus: ([?], $\square$ ) Directrix: $x=$ $\square$
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Solution

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Solution Steps

Step 1: Identify the Form of the Parabola

Given the parabola in the form \((y-k)^2 = 4p(x-h)\), we identify it opens left or right.

Step 2: Calculate the Focus

The focus is calculated using the formula \((h+p, k)\). Given \(h=1\), \(k=1\), and \(p=1\), the focus is at \((2, 1)\).

Step 3: Calculate the Directrix

The directrix is a vertical line, calculated using the formula \(x = h-p\). Given \(h=1\) and \(p=1\), the directrix is \(x = 0\).

Final Answer:

The focus of the parabola is at (2, 1), and the directrix is at x = 0.

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