Question

question 5 (multiple choice worth 1 points)
(07.03 hc)
right triangle abc is located at a (-1, 4), b (-1, 1), and c (-5, 1) on a coordinate plane. what is the equation of a circle a with radius ac?
\\(\circ (x + 1)^2 + (y - 4)^2 = 9\\)
\\(\circ (x + 5)^2 + (y - 1)^2 = 25\\)
\\(\circ (x + 5)^2 + (y - 1)^2 = 16\\)
\\(\circ (x + 1)^2 + (y - 4)^2 = 25\\)

Explanation:

Step1: Find the length of AC (radius)

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For points \(A(-1,4)\) and \(C(-5,1)\):
\(x_1=-1,y_1 = 4,x_2=-5,y_2 = 1\)
\(AC=\sqrt{(-5 - (-1))^2+(1 - 4)^2}=\sqrt{(-4)^2+(-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5\)
So the radius \(r = 5\).

Step2: Find the center of the circle

The center of the circle is point \(A(-1,4)\), so \(h=-1,k = 4\) (the standard form of a circle is \((x - h)^2+(y - k)^2=r^2\), which can be rewritten as \((x+1)^2+(y - 4)^2=r^2\) since \(h=-1\)).

Step3: Substitute \(r\) into the circle equation

We found \(r = 5\), so \(r^2=25\). The equation of the circle is \((x + 1)^2+(y - 4)^2=25\) (Wait, wait, no, wait. Wait, let's check the options again. Wait, maybe I made a mistake. Wait, the options have \((x + 1)^2+(y - 4)^2=9\), \((x + 5)^2+(y - 1)^2=25\), \((x + 5)^2+(y - 1)^2=16\), \((x + 1)^2+(y - 4)^2=25\). Wait, no, wait, the center is \(A(-1,4)\), radius is \(AC = 5\), so the equation should be \((x+1)^2+(y - 4)^2=25\), which is the last option? Wait, no, wait the options:

Wait the options are:

1. \((x + 1)^2+(y - 4)^2=9\)

1. \((x + 5)^2+(y - 1)^2=25\)

1. \((x + 5)^2+(y - 1)^2=16\)

1. \((x + 1)^2+(y - 4)^2=25\)

Wait, let's recalculate \(AC\) again. \(A(-1,4)\), \(C(-5,1)\):

\(\Delta x=-5-(-1)=-4\), \(\Delta y=1 - 4=-3\)

\(AC=\sqrt{(-4)^2+(-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5\), so radius \(r = 5\), center \((-1,4)\), so equation is \((x + 1)^2+(y - 4)^2=25\), which is the fourth option. Wait, but let's check the options again. Wait the fourth option is \((x + 1)^2+(y - 4)^2=25\). So that's the correct one.

Wait, maybe I misread the options earlier. So the correct equation is \((x + 1)^2+(y - 4)^2=25\) (the fourth option).

Answer:

\((x + 1)^2+(y - 4)^2=25\) (the option: \((x + 1)^2+(y - 4)^2=25\))