Question

ex. 4) (y - 1)^2 = -4(x - 1) v(1,1) general form of a parabola

Explanation:

Step1: Identify the form of parabola

The equation \((y - 1)^2=-4(x - 1)\) is of the form \((y - k)^2 = 4p(x - h)\), where \((h,k)\) is the vertex of the parabola. Here \(h = 1,k = 1\), so the vertex \(V=(1,1)\).

Step2: Determine the value of \(p\)

Comparing \((y - 1)^2=-4(x - 1)\) with \((y - k)^2 = 4p(x - h)\), we have \(4p=-4\), so \(p=- 1\).

Step3: Find the focus

For a parabola of the form \((y - k)^2 = 4p(x - h)\), the focus is given by the point \((h + p,k)\). Substituting \(h = 1,k = 1,p=-1\), we get the focus \(F=(1+( - 1),1)=(0,1)\).

Step4: Find the directrix

The directrix of a parabola of the form \((y - k)^2 = 4p(x - h)\) is the line \(x=h - p\). Substituting \(h = 1,p=-1\), we get \(x=1-(-1)=2\), so the directrix is \(x = 2\).

Answer:

Vertex: \((1,1)\), Focus: \((0,1)\), Directrix: \(x = 2\)